Vector Wave Equation Research Articles

Complete classification of stationary flows with constant total pressure of ideal incompressible infinitely conducting fluid

The exhaustive classification of stationary incompressible flows with constant total pressure of ideal infinitely electrically conducting fluid is given. By introduction of curvilinear coordinates based on streamlines and magnetic lines of the flow, the system of magnetohydrodynamics equations is reduced to a nonlinear vector wave equation extended by the incompressibility condition in a form of a generalized Cauchy integral. For flows with constant total pressure, the wave equation is explicitly integrated, whereas the incompressibility condition is reduced to a scalar equation for functions, depending on different sets of variables. The central difficulty of the investigation is the separation of variables in the scalar equation, and integration of the resulting overdetermined systems of nonlinear partially differential equations. The canonical representatives of all possible types of solutions together with equivalence transformations that extend the canonical set to the whole amount of solutions are represented.

Simulation of electromagnetic scattering by random discrete particles using a hybrid FE-BI-CBFM technique

A hybrid finite element–boundary integral–characteristic basis function method (FE-BI-CBFM) is proposed for an efficient simulation of electromagnetic scattering by random discrete particles. Specifically, the finite element method (FEM) is used to obtain the solution of the vector wave equation inside each particle and the boundary integral equation (BIE) using Green's functions is applied on the surfaces of all the particles as a global boundary condition. The coupling system of equations is solved by employing the characteristic basis function method (CBFM) based on the use of macro-basis functions constructed according to the Foldy–Lax multiple scattering equations. Due to the flexibility of FEM, the proposed hybrid technique can easily deal with the problems of multiple scattering by randomly distributed inhomogeneous particles that are often beyond the scope of traditional numerical methods. Some numerical examples are presented to demonstrate the validity and capability of the proposed method.

Scattering of arbitrarily incident Gaussian beams by fractal soot aggregates

A hybridization of the finite element and boundary integral methods is applied to the study of light scattering by fractal soot aggregates illuminated by Gaussian beams with arbitrary incidence. In particular, the Davis–Barton fifth-order approximation in combination with rotation Euler angles is employed to represent the arbitrarily incident Gaussian beams. The finite element method is used to obtain the solution of the vector wave equation inside each primary particle and the boundary integral equations are applied on the surfaces of all the particles as a global boundary condition. The resultant matrix equation is solved by an iterative method, where the multilevel fast multipole algorithm can be employed to speed up the matrix–vector multiplication. Some numerical results are included to illustrate the validity of the present method and to show the scattering behaviors of fractal soot aggregates when they are illuminated by Gaussian beams.

The element level time domain (ELTD) method for the analysis of nano-optical systems: I. Nondispersive media

We study a 3-dimensional, dual-field, fully explicit method for the solution of Maxwell's equations in the time domain on unstructured, tetrahedral grids. The algorithm uses the element level time domain (ELTD) discretization of the electric and magnetic vector wave equations. In particular, the suitability of the method for the numerical analysis of nanometer structured systems in the optical region of the electromagnetic spectrum is investigated. The details of the theory and its implementation as a computer code are introduced and its convergence behavior as well as conditions for stable time domain integration is examined. Here, we restrict ourselves to non-dispersive dielectric material properties since dielectric dispersion will be treated in a subsequent paper. Analytically solvable problems are analyzed in order to benchmark the method. Eventually, a dielectric microlens is considered to demonstrate the potential of the method. A flexible method of 2nd order accuracy is obtained that is applicable to a wide range of nano-optical configurations and can be a serious competitor to more conventional finite difference time domain schemes which operate only on hexahedral grids. The ELTD scheme can resolve geometries with a wide span of characteristic length scales and with the appropriate level of detail, using small tetrahedra where delicate, physically relevant details must be modeled.

Scattering in multilayered structures: Diffraction from a nanohole

The spectral expansion of the Green's tensor for a planar multilayered structure allows us to semi analytically obtain the angular spectrum representation of the field scattered by an arbitrary dielectric perturbation present in the structure. In this paper we present a method to find the expansion coefficients of the scattered field, given that the electric field inside the perturbation is available. The method uses a complete set of orthogonal vector wave functions to solve the structure's vector wave equation. In the two semi-infinite bottom and top media, those vector wave functions coincide with the plane-wave basis vectors, including both propagating and evanescent components. The technique is used to obtain the complete angular spectrum of the field scattered by a nanohole in a metallic film under Gaussian illumination. We also show how the obtained formalism can easily be extended to spherically and cylindrically multilayered media. In those cases, the expansion coefficients would multiply the spherical and cylindrical vector wave functions.

Sensitivity Analysis of Microwave Circuits Using Parameterized Model Order Reduction Techniques

This paper presents an efficient technique to calculate sensitivities of microwave structures with respect to network design parameters. The proposed technique uses a parametric reduced order model derived from the finite element method discretization of the vector wave equation to solve the original network and an adjoint variable method to calculate sensitivities. An important feature of the proposed method is that the solution of the original network as well as sensitivities with respect to any parameter is obtained from the solution of the reduced order model.

Study of Self Phase Modulation in Chalcogenide Glass Photonic Crystal Fiber

In this paper, Self Phase Modulation (SPM) in chalcogenide As2Se3 glass Photonic Crystal Fiber (PCF) is numerically studied by combining the fully vectorial effective index method (FVEIM) and Split Step Fourier Method (SSFM). The FVEIM is used to calculate the variation of effective refractive index of guided mode (neff), effective area (Aeff), dispersion and non-linear coefficient (γ) with wavelength for different designs of chalcogenide As2Se3 PCF. The FVEIM solves the vector wave equations and SSFM solves non linear Schrödinger Equation (NLSE) for the different designing parameter of As2Se3 PCF. In case of Self Phase Modulation (SPM), spectral width of the obtained output pulse at d/Λ=0.7 is 1.5 times greater than width of the output pulse obtained at d/L=0.3 using SSFM. Thus we can get the desired spectral broadening just by tailoring the design parameters of the PCF.

Transient analysis of microwave Gunn oscillator using extended spectral element time domain method

In this study, the microwave Gunn oscillator is analyzed by a hybrid electromagnetic circuit simulator, which is based on the spectral element time domain (SETD) method. The Gunn diode within the oscillator is treated as a lumped element, while the passive distributed part of the oscillator is modeled using the SETD method. In order to incorporate the contribution of the Gunn diode into the SETD context, the SETD method is extended by introducing a lumped current term into the second‐order vector wave equation. When Galerkin's method is used for the space discretization and the central difference scheme is used for time stepping, a global SETD system involving the Gunn diode is assembled. Furthermore, the global system matrix is block‐diagonal and the inversion of this matrix can be easily implemented, and thus the extended SETD method is a fully explicit solver and CPU time can be significantly reduced. By virtue of this method, the strong nonlinear feature of the Gunn oscillator is well characterized in the time domain, such as the phenomenon of injection locking. Numerical results demonstrate the ability and effectiveness of the extended SETD method for the fast analysis of microwave Gunn oscillator.

Numerical simulation of focused nonlinear acoustic beams: A Fourier continuation direct solver and comparisons to a paraxial approximation.

A recently developed Fourier-continuation (FC) method is used to develop a numerical algorithm, which can accurately solve the fully nonlinear acoustic vector wave equation for the type of hundred-wavelength domains arising in the field of focused medical ultrasound. The FC nonlinear-ultrasound solver is used to outline the domain of applicability of the widely used Khokhlov–Zabolotskaya–Kuznetsov (KZK) approximation in focused ultrasound particularly in configurations arising in the field of high intensity focused ultrasound (HIFU). Good agreement was found for small-radius transducers; but as the source radius was increased to dimensions equivalent to practical HIFU sources the KZK approximation leads to errors in the location of the predicted focus. Examples for media with multiple scattering centers also show the utility of the FC approach in modeling complex media. In general the FC method produces solutions with significantly reduced discretization requirements over those associated with finite-difference and finite-element methods, and they lead to improvements in computing times by factors of hundreds and to thousands over competing approaches. [Work supported by NSF Grant No. 0835795.]

Magnetic Resonance Driven Electrical Impedance Tomography: A Simulation Study

Magnetic resonance electrical impedance tomography (MREIT) is a method for reconstructing a three-dimensional image of the conductivity distribution in a target volume using magnetic resonance (MR). In MREIT, currents are applied to the volume through surface electrodes and their effects on the MR induced magnetic fields are analyzed to produce the conductance image. However, current injection through surface electrodes poses technical problems such as the limitation on the safely applicable currents. In this paper, we present a new method called magnetic resonance driven electrical impedance tomography (MRDEIT), where the magnetic resonance in each voxel is used as the applied magnetic field source, and the resultant electromagnetic field is measured through surface electrodes or radio-frequency (RF) detectors placed near the surface. Because the applied magnetic field is at the RF frequency and eddy currents are the integral components in the method, a vector wave equation for the electric field is used as the basis of the analysis instead of a quasi-static approximation. Using computer simulations, it is shown that complex permittivity images can be reconstructed using MRDEIT, but that improvements in signal detection are necessary for detecting moderate complex permittivity changes.

Natural curvilinear coordinates for ideal MHD equations. Non-stationary flows with constant total pressure

Equations of magnetohydrodynamics (MHD) in the natural curvilinear system of coordinates where trajectories and magnetic lines play a role of coordinate curves are reduced to the non-linear vector wave equation coupled with the incompressibility condition in the form of the generalized Cauchy integral. The symmetry group of obtained equation, equivalence transformation, and group classification with respect to the constitutive equation are calculated. New exact solutions with functional arbitrariness describing non-stationary incompressible flows with constant total pressure are given by explicit formulae. The corresponding magnetic surfaces have the shape of deformed nested cylinders, tori, or knotted tubes.

Combined Electromagnetic Energy and Momentum Conservation Equation

We have combined the two fundamental conservation theorems in electromagnetic theory-the energy and the momentum conservation theorems-to derive a vector wave equation for linear and homogeneous media with symmetrical constitutive matrix. We have demonstrated how the relation between the Poynting vector and the momentum density vector affects the electromagnetic quantities. Furthermore, a relation between the Poynting vector, the momentum density vector, and the phase velocity of the Poynting vector has been derived. In addition, what happens if the Poynting vector and the momentum density vector become mutually orthogonal or one of them vanishes has also been discussed. The angle between the above-mentioned vectors has been derived, and a relation between the momentum density vector and the wave vector for plane waves has been obtained. It has been shown that for plane waves, orthogonality of the phase velocity is not feasible.

Higher order modes of coupled optical fibres

The structure of hybrid higher order modes of two coupled weakly guiding identicaloptical fibres is studied. On the basis of perturbation theory with degeneracy forthe vector wave equation expressions for modes with azimuthal angular numberl ≥ 1 are obtained that allow for the spin–orbit interaction. The spectra of polarizationcorrections to the scalar propagation constants are calculated in a wide range of distancesbetween the fibres. The limiting cases of widely and closely spaced fibres arestudied. The obtained results can be used for studying the tunnelling of opticalvortices in directional couplers and in matters concerned with information security.

Application of the Sh-matrices method to light scattering by spheroids

The application of the Sh-matrices technique developed within theT-matrix formalism allows for the derivation of analytic solutions to the light scatteringfrom non-spherical particles. In this paper we derive such solutions for isolatedprolate and oblate spheroids. Examples of calculations of spheroid scatteringproperties including the case of particles with high eccentricity are presented. Ourresults are compared with corresponding calculations using the discrete dipoleapproximation (DDA) method and analytical methods based on the separationof variables for the vector wave equations in the spheroidal coordinate system.

Full Wave Analyses of Electromagnetic Fields With an Iterative Domain Decomposition Method

This paper describes large-scale full wave analyses of electromagnetic fields by the finite-element method with an iterative domain decomposition method (IDDM). A stationary vector wave equation for the high-frequency electromagnetic field analyses is solved taking an electric field as an unknown function. Then, to solve subdomain problems by the direct method, the direct method based on the LDLT decomposition method is introduced in subdomains. If the direct method is applied for solving subdomain problems, the computation time seems to be reduced by the improved accuracy of subdomain problems, and storing matrices that are results of the decomposition on main memory.

Application of the shifted-Laplace preconditioner for iterative solution of a higher order finite element discretisation of the vector wave equation: First experiences

The analysis of electromagnetic scattering by electrically large one-side open–ended cavities remains a challenge. Finite element discretisation of the vector wave equation to solve for the electric field inside the cavity leads to ill-conditioned indefinite linear systems of large dimension, a result of the requirement of a fine nearly uniform discrete sampling in the computational domain. Direct methods based on frontal solution techniques to solve the resulting linear system have been used because efficient iterative methods were not available. With the arrival of the shifted-Laplace preconditioner of Erlangga, iterative solution of the indefinite system becomes tractable. This paper discusses the modifications required for application of the shifted-Laplace preconditioner to cavity scattering and some preliminary results of this approach. The shifted-Laplace preconditioner is shown to be very effective for improving the convergence rate of the iterative solution algorithm. However, to be able to handle problems with larger number of degrees of freedom, it is necessary to include a multigrid algorithm to solve the preconditioner system, as this will allow the use of short recurrence Krylov subspace methods, as opposed to the currently employed long-recurrence method, for which the storage requirements of the Krylov basis become unpractically large.

Analysis of photonic crystal defect modes by maximal symmetrization and reduction

We analyze in depth the eigenmodes symmetry of the vectorial electromagnetic wave equation with discrete symmetry, using a recently developed maximal symmetrization and reduction scheme leading to an automatic technique which decomposes every mode into its most fundamental internal geometrical components carrying independent symmetries, the ultimately reduced component functions (URCFs). Using URCFs, geometrical properties of photonic crystal defect modes can be analyzed in great details. In particular we analytically identify the kind of modes that display non-vanishing transverse electric or transverse magnetic amplitude at the cavity center in C2v, C3v, C4v, and C6v symmetries, and their degeneracies. We also build a postprocessing tool able to extract and identify URCFs out of the modes whether from experimental or numerical origin. In the latter case it is independent of the eigenmode computation method. In another variant the whole eigenmode computation can be systematically reduced to a minimal domain, without any need for applying specific non-trivial boundary conditions. The approach leads to strong analytical predictions which are illustrated for specific H1 and L3 cavities using the postprocessing tool on full three-dimensional computed modes. It not only constitutes an unprecedented check of the symmetry of the computational results, but it is shown to also deliver a deep geometrical and physical insight into the structure of the modes of photonic bandgap microcavities, which is of direct use for most modern applications in quantum photonics.

Bosons, fermions and anyons in the plane, and supersymmetry

Universal vector wave equations allowing for a unified description of anyons, and also of usual bosons and fermions in the plane are proposed. The existence of two essentially different types of anyons, based on unitary and also on non-unitary infinite-dimensional half-bounded representations of the (2 + 1)D Lorentz algebra is revealed. Those associated with non-unitary representations interpolate between bosons and fermions. The extended formulation of the theory includes the previously known Jackiw–Nair (JN) and Majorana–Dirac (MD) descriptions of anyons as particular cases, and allows us to compose bosons and fermions from entangled anyons. The theory admits a simple supersymmetric generalization, in which the JN and MD systems are unified in N = 1 and N = 2 supermultiplets. Two different non-relativistic limits of the theory are investigated. The usual one generalizes Lévy–Leblond 's spin 1/2 theory to arbitrary spin, as well as to anyons. The second, “Jackiw–Nair” limit (that corresponds to Inönü–Wigner contraction with both anyon spin and light velocity going to infinity), is generalized to boson/fermion fields and interpolating anyons. The resulting exotic Galilei symmetry is studied in both the non-supersymmetric and supersymmetric cases.

Design of 3-D Periodic Metamaterials for Electromagnetic Properties

This paper presents a new homogenization formula to compute the effective electromagnetic properties for periodic metamaterials. Numerical examples showed that the effective permittivity and permeability of the composites with cubic inclusions, formerly known to have the lowest permittivity, are closer to the Hashin-Strikman bounds than those obtained from other methods. To tailor the specific effective properties, an inverse homogenization procedure is proposed within the framework of vector wave equations. Some novel metamaterial microstructures with a range of specific effective permittivity and/or permeability are obtained. By maximizing the permittivity and permeability at the same time, a structure with minimal surface area (the mean curvature of the surface equals zero everywhere), namely, the well-known Schwarz primitive structure, is obtained. Similarly to the nano-spheres (dielectric spheres covered by plasmonic shells) with negative refraction, we generalize the Schwarz primitive structure and its analogy (e.g., those with a constant mean curvature surface) to one class of chiral composites by embedding one of these structures with smaller volume fraction (nonmagnetic inclusive cores) into another with large volume fraction (metal shell). Such composites have potential to provide better behaviors because they can best utilize different components. The anisotropic composites and multiple solutions to the inverse homogenization are also illustrated.

An Investigation of a Preconditioner for High-Frequency Maxwell Problems

This article describes a preconditioner for the indefinite linear equation system that is obtained when the vector wave equation is discretized by hp edge finite elements. A theoretical investigation of the eigenspectrum of the preconditioned system for p = 1 quadrilateral elements illustrates that the preconditioner will perform well, once the mesh spacing is sufficiently fine. Numerical experiments are included, which demonstrate the effectiveness of the preconditioner for h and p refinements in both two and three dimensions.