Vector Wave Equation Research Articles

Modal Characteristics of Photonic Crystal Fibers

The modal characteristics of the photonic crystal fibers are analyzed using the reliable and efficient plane wave expansion method. The mode profile, effective index and group velocity dispersion are obtained by solving Maxwell's vector wave equations without any approximation. The zero dispersion condition of a photonic crystal fiber is derived over a wide range of wavelengths. Higher-order modes are also easily found as a by-product of the plane wave expansion method. This method can be used to quickly and accurately design various optical properties of photonic crystal fibers.

Propagation of the transient electromagnetic field above atmospheric surface duct

The transient electromagnetic field generated by a vertical magnetic dipole above an evaporation duct is investigated theoretically. The source of the electromagnetic field is taken to be a vertical magnetic dipole above an evaporation duct with arbitrary time varying moment. The application of two integral transform to the wave equation for the Fitzgerald vector––a Laplace transform in time and a two-dimensional Fourier transform in the horizontal coordinates in space-leads, under consideration of initial boundary and transition condition. The modified Cagniard method is used to derive closed form expressions for Fitzgerald vector anywhere above the duct and based on evaluating the field in a series of image sources of the primary source. Hence we can give a physically intuitive description of the polarization dependence of the time.

A comprehensive VCSEL device simulator

This paper deals with the design and implementation of a self-consistent electrothermooptical device simulator for vertical-cavity surface-emitting lasers (VCSELs). The model is based on the photon rate equation approach. For the bulk electrothermal transport, a thermodynamic model is employed in a rotationally symmetric body. Heterojunctions are modeled using a thermionic emission model and quantum wells are treated as scattering centers for carriers. The optical field is expanded into modes that are eigensolutions of the vectorial electromagnetic wave equation with an arbitrary, complex dielectric function. The open nature of the VCSEL cavity is treated by employing perfectly matched layers. The optical gain and absorption model in the quantum-well active region is based on Fermi's Golden Rule. The subbands in the quantum well are determined by solving the stationary Schrodinger equation and using a parabolic band approximation for the electrons, light and heavy holes. The photon rate equation is fully integrated into the Newton-Raphson scheme used to solve the system of nonlinear device equations. An efficient numerical optical mode solver is used, that is based on a Jacobi-Davidson type iterative eigensolver. The latter combines a continuation scheme with preconditioner recycling. The practical relevance of the implementation is demonstrated with the simulation of a realistic etched-mesa VCSEL device.

A stabilized multilevel vector finite-element solver for time-harmonic electromagnetic waves

An enhanced finite-element method (FEM) for the vector wave equation is presented. For improved speed and stability ranging from microwave frequencies down to the static limit, we propose a multilevel solver that uses a tree-gauged formulation on the coarsest mesh and a partially gauged scheme for the iterative cycle. Moreover, we have generalized the concept of hanging nodes to higher order H(curl)-conforming tetrahedral elements. The combination of hierarchical basis functions and the hanging variables framework yields great flexibility in placing degrees of freedom and provides a very attractive alternative to remeshing in an hp-adaptive context.

Ray-based methods in multidimensional linear wave conversion

A tutorial introduction to the topic of linear wave conversion in multiple spatial dimensions is provided. The emphasis is on physical concepts, particularly those features of multidimensional conversion that are new and different from the more familiar “mode conversion” problem in one spatial dimension. After introductory comments, a brief review of WKB theory for vector wave equations in the absence of conversion is provided in order to introduce notation, terminology, and geometrical ideas. A primary theme of the discussion is that, although WKB (ray-based) methods break down in conversion regions, the ray geometry in the conversion region can be used to develop local wave equations that govern the two coupled wave channels undergoing conversion. These methods can be incorporated into ray-tracing algorithms providing, for the first time, the ability to follow the “ray splitting” associated with linear conversion in multidimensions, including the amplitude and phase changes associated with the conversion.

Adjoint high-order vectorial finite elements for nonsymmetric transversally anisotropic waveguides

In this paper, a vector finite-element method (FEM) for the investigation of microwave and optical waveguides with arbitrary cross section is presented. In particular, the FEM solves the vector wave equation in the frequency domain for waveguides, containing nonsymmetric transversally anisotropic (passive and active) materials. Therefore, a bilinear form has been derived that is solved for the original and adjoint wave equation. Biorthogonality, relations of original and adjoint solutions for simpler materials, the role of nonphysical solutions for both problems, and their proper approximations are discussed in detail. Special focus has been set on shifting the propagation constants of the nonphysical modes for nonzero frequencies to infinity. Therefore, an excellent rate of convergence for iterative solvers is observed. The occurrence of nonphysical solutions is shown. Different nth-order basis functions are systematically derived, which are identical to the well-known tangential continuous finite elements. Both algebraic eigenvalue problems are solved via a generalized Lanczos algorithm, which solves both eigenvalue problems efficiently. The capability of the FEM is illustrated by critical examples.

Closed-form eigenfrequencies in prolate spheroidal conducting cavity

In this paper, an efficient approach is proposed to analyze the interior boundary-value problem in a spheroidal cavity with perfectly conducting wall. Since the vector wave equations are not fully separable in spheroidal coordinates, it becomes necessary to double-check validity of the vector wave functions employed in analysis of the vector boundary problems. In this paper, a closed-form solution has been obtained for the eigenfrequencies f/sub ns0/ based on TE and TM cases. From a series of numerical solutions for these eigenfrequencies, it is observed that the f/sub ns0/ varies with the parameter /spl xi/ among the spheroidal coordinates (/spl eta/, /spl xi/, /spl phi/) in the form of f/sub ns0/(/spl xi/) =f/sub ns/(0)[1+g/sup (1)///spl xi//sup 2/+g/sup (2)///spl xi//sup 4/+g/sup (3)///spl xi//sup 6/+/spl middot//spl middot//spl middot/]. By means of the least squares fitting technique, the values of the coefficients, g/sup (1)/, g/sup (2)/, g/sup (3)/, ..., are determined numerically. It provides analytical results and fast computations of the eigenfrequencies, and the results are valid if /spl xi/ is large (e.g., /spl xi//spl ges/100).

Time-domain finite-element simulation of three-dimensional scattering and radiation problems using perfectly matched layers

An effective algorithm to construct perfectly matched layers (PMLs) for truncating time-domain finite-element meshes used in the simulation of three-dimensional (3-D) open-region electromagnetic scattering and radiation problems is presented. Both total- and scattered-field formulations are described. The proposed algorithm is based on the time-domain finite-element solution of the vector wave equation in an anisotropic and dispersive medium. The algorithm allows for the variation of the PML parameters within each element, which facilitates the efficient use of higher order vector basis functions. The stability of the resultant numerical procedure is analyzed, and it is shown that unconditionally stable schemes can be obtained. Numerical simulations of radiation and scattering problems based on both the zeroth- and higher order vector bases are presented to validate the proposed PML scheme.

IMAGING OF HELICAL SURFACE WAVE MODES IN THE NEAR FIELD

A K-space technique that has proven successful for analyzing acoustic fields is applied to electromagnetic fields in the very near reactive region. In cylindrical coordinates the K- space spectrum describes helical wave modes on a fixed radius cylindrical surface that represent either propagating and evanescent mode energy. When this surface encloses a radiating or scattering object the resulting K-space decomposition reveals information about the physics of the wave-object interaction. Originally applied to the scalar wave equations of acoustics this method has been adapted to the vector wave equations encountered in electromagnetics. Here, the K-space decomposition is applied to the near electric and magnetic fields generated by a plane wave scattered from a perfectly conducting, finite length cylinder. We numerically simulate electromagnetic fields scattered from cylinders with a smooth surface and a surface with periodic grooves to illustrate the ability of the K-space method to differentiate between propagating and evanescent modal energy. Features in the K-space spectrum that impact the far field bi-static scattering cross section are identified to provide the link between the near field analysis and far field cross section.

Three-dimensional orthogonal vector basis functions for time-domain finite element solution of vector wave equations

Three-dimensional (3-D) orthogonal vector basis functions are developed for the time-domain finite element solution of vector wave equations. These basis functions enforce both the tangential continuity of the electric field and the normal continuity of the electric flux. The stability of the resulting time-domain finite element schemes is investigated and demonstrated to be guaranteed. The use of the proposed basis functions completely eliminates the matrix solution at each time step required by the time-domain finite element solution of vector wave equations. The computational cost thereby scales as O(N/sub t/N) with N/sub t/ and N denoting the number of time steps and the number of unknowns, respectively. Defined over tetrahedral elements, the proposed basis functions increase the solution efficiency without compromising the geometry modeling flexibility. Both numerical results and comparison with traditional vector basis functions demonstrate the accuracy as well as the efficiency of the proposed three-dimensional orthogonal vector bases.

An effective algorithm for implementing perfectly matched layers in time-domain finite-element simulation of open-region EM problems

An algorithm is presented to implement perfectly matched layers (PMLs) for the time-domain finite-element (TDFE) simulation of two-dimensional open-region electromagnetic scattering and radiation problems. The proposed algorithm is based on the TDFE solution of a special vector wave equation similar to the one in an anisotropic and dispersive medium. The impact of the PML on the stability of the resultant TDFE solution is studied for a variety of temporal discretization schemes, and it is shown that the proposed algorithm for implementing PML can support unconditionally stable TDFE schemes. Both the total- and the scattered-field formulations are described, and numerical simulations of radiation and scattering problems are presented to validate the proposed PML algorithm for the mesh truncation of the TDFE solution.

The solution of electromagnetic fields by equivalent magnetical network

A simple and explicit derivation is given for the analyzed electromagnetic fields by an equivalent magnetical network. By the circuit-oriented finite element method the vector wave equation problem is transformed in an algebraical system and solved. The original method is theoretically formulated and is applied in test configurations. The results of the simulations are then compared with those obtained by the traditional edge element finite element method solution.

Numerical simulation of first-order backscattered P and S waves for time-lapse seismic imaging in heterogeneous reservoirs

Summary First-order elastic wave scattering theory, validated in general features by deep-borehole seismic coda wave observations, is used to numerically simulate time-lapse borehole seismic data aimed at monitoring oil–water substitution in heterogeneous hydrocarbon reservoirs. A first-order perturbation solution of the vector wave equation leads to P–P and S–S backscattered displacement vector motion expressed as the angular summation of the second radial derivative of P- and S-wave velocity and density fluctuations α′/α, β′/β, ρ′/ρ over expanding spherical wave fronts of radii ξ = αt/2 and ξ = βt/2. P- and S-wave velocity and density spatial heterogeneity is modelled as long-range correlated random fluctuations consistent with the power-law-scaling character of crustal rock physical properties measured by borehole logs. The spectra of model scattering displacement seismograms for a power-law-scaling volumetric noise distribution duplicate the frequency enrichment observed in the spectra of broadband earthquake coda waves recorded in a deep well. Modelling of time-lapse scattering in power-law-scaling permeability structures suggests that a stable borehole seismic source can locate oil–water substitution in reservoir volumes tens of metres on a side at distances of up to a few hundred metres from an observation well using currently available borehole seismic technology. Time-lapse tracking of oil–water substitution and the monitoring of reservoir stress conditions can lead to spatially well-constrained reservoir models, despite large-scale, large-amplitude correlated random heterogeneity.

Conformal mapping methods for variable parameter elastodynamics

This work is a generalization of earlier efforts to develop free-space fundamental solutions (Green’s functions) for a certain class of non-homogenous materials (i.e., materials with position-dependent elastic parameters and density) under time-harmonic conditions by means of conformal mapping methods. These methods were shown to be very effective in conjunction with the scalar wave equation, but their extension to the 2D and 3D vector wave equations of elastodynamics is hampered by the following two factors: (a) the governing equations of motion are vectorial, which implies that any transformation of coordinates affects the basis vectors used in the original (i.e., Cartesian) coordinate system; and (b) an extension of conformal mapping to a 3D coordinate system is not readily obvious. In the present work, both issues raised above are addressed. Furthermore, the necessary background is established whereby the construction of Green’s functions for 2D and 3D elastodynamics can be systematically accomplished in a mapped space, followed by an inversion to the original Cartesian coordinate system. Of course, conformal mapping as used here is inherently an inverse method, in the sense that the material parameter profiles for a given problem are recovered as constraints during the solution procedure and therefore depend on the type of mapping used.

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.

The solution of electromagnetic fields by equivalent magnetic network

A simple and explicit derivation is given for the analyzed electromagnetic fields by an equivalent magnetic network. By the circuit-oriented finite element method the vector wave equation problem is transformed into an algebraical system and solved. The original method is theoretically formulated and applied in test configurations. The results of the simulations are then compared with those obtained by the traditional edge element finite element method solution.

Comprehensive modeling of vertical-cavity laser-diodes by the method of lines

A comprehensive numerical model for the analysis of vertical-surface emitting-lasers is presented. An optical, electrical, and thermal submodel are introduced. The complete analysis is based on the method of lines. The temperature distribution and the current density are calculated in the whole structure. The optical behavior is investigated with a full vectorial wave equation in cylindrical coordinates. Multimode effects are considered when calculating the optical output power. A rate equation for electrons and holes is used, which includes diffusion and recombination effects inside the quantum well. By combining all submodels a self consistent solution is found.

Time-domain finite-element modeling of dispersive media

A general formulation is described for time-domain finite-element modeling of electromagnetic fields in a general dispersive medium. The formulation is based on the second-order vector wave equation and incorporates the dispersion effect of a medium via a recursively evaluated convolution integral. This evaluation is kept to second order in accuracy using linear interpolation within each time step. Numerical examples are given to validate the proposed formulation.

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.

Acoustic chaos

We use high resolution acoustic spectroscopy as an analog system to quantum systems. We study transitions between regularity and chaos and specific symmetry breaking. We compare with the predictions of Random Matrix Theory that was developed for the study of the complicated compound nuclear states. The acoustic system is interesting in its own right. It is governed by a vectorial wave-equation in contrast to the scalar Schrödinger equation, and we study the complex interplay of different resonant modes arising from this fact.