Helmholtz Wave Equation Research Articles

Quadridirectional eigenmode expansion scheme for 2-D modeling of wave propagation in integrated optics

The propagation of guided and nonconfined optical waves at fixed frequency through dielectric structures with piecewise constant, rectangular permittivity is considered in two spatial dimensions. Bidirectional versions of eigenmodes, computed for sequences of multilayer slab waveguides, constitute the expansion basis for the optical electromagnetic field. Dirichlet boundary conditions are sufficient to discretize the mode sets. Superpositions of two such expansions (bidirectional eigenmode propagation (BEP) fields), oriented along the two perpendicular coordinate axes, establish rigorous semianalytical solutions of the relevant Helmholtz wave equation on an unbounded, cross-shaped computational domain. The overlap of the lateral windows of the two BEP sets can be viewed as a rectangular computational window with fully transparent boundaries. Simulation results for a series of model systems (Gaussian beams in free space, Bragg gratings, waveguide crossings, a square cavity with perpendicular ports, and a 90° bend in a photonic crystal waveguide) illustrate the performance of the approach.

Bounds on Extrapolation of Field Knowledge for Long-Range Prediction of Mobile Signals

A bound is derived for the extent to which narrowband knowledge of a multipath field inside a spatially limited region can be extrapolated beyond that region. The bound is independent of any model which may be applied to the multipath field, relying on the field being a solution to the homogeneous Helmholtz wave equation. The bound has implications for the extent and reliability of long-range channel prediction, particularly when the multipath is diffuse.

Cut-off frequency variation in optical waveguides due to photon tunnelling

We show that the cut-off frequency (low corner frequency) for an undersized waveguide can change due to the photon tunnelling phenomenon. The Helmholtz wave equation will be used to calculate the tunnelling time of a photon through a refraction index barrier. We report an analytical expression for the transit time of the photon in a refraction index barrier as well as the transmission coefficient. It is predicted that the tunnelling time depends on the refraction index, the thickness of the barrier and also on the incident wave frequency. We will exploit this idea to obtain light coupling to a classically forbidden region, hence describing the cut-off frequency variation. A comparison with the experimental data is also reported.

Fast multipole accelerated boundary integral equation methods

Fundamentals of Fast Multipole Method (FMM) and FMM accelerated Boundary Integral Equation Method (BIEM) are presented. Developments of FMM accelerated BIEM in the Laplace and Helmholtz equations, wave equation, and heat equation are reviewed. Applications of these methods in computational mechanics are surveyed. This review article contains 173 references.

The initiation, termination, and evolution of nonparaxial optical vortices: High-order singular beams

Exact solutions to the scalar Helmholtz wave equation describing nonparaxial high-order mode beams are considered. To match these solutions to those of the parabolic wave equation, soft approximate boundary conditions are formulated. According to these conditions, in the paraxial limit kz 0≫1, paraxial and nonparaxial wave beams transform into each other either far off the focal caustic (at z≫z 0) or near it. The behavior of the mode beams with different azimuthal and radial mode indices is described.

Electromagnetic fields of thin circular loop antenna of arbitrary radius

The radiation pattern of a circular loop source of alternating current located in the lower half-space of two conductive media is considered. A solution of the linear Maxwell's equations in the frequency domain, using the Hankel transform to modified Helmholtz wave equations, was obtained. This leads to a system of ordinary differential equations with respect to the vertical coordinate, considering initial, boundary, and transition conditions. The solution determines the electromagnetic field in the conductive media which consists of primary and scattered waves. The results obtained from this paper can be used to evaluate numerical solutions of different complexities such as the underlying multilayered media. In addition, the derived formulation would be useful for interpreting airborne electromagnetic systems. PACS Nos.: 41.20Jb, 42.25Bs, 42.25Gy, 44.05+e

An efficient method to analyze the H-plane waveguide junction circulator with a ferrite sphere

This paper presents an approximate, but efficient field treatment of the new easy-to-fabricate ferrite-sphere-based H-plane waveguide circulator for potentially low-cost millimeter-wave communication systems. A new three-dimensional modeling strategy using a self-inconsistent mixed-coordinates-based mode-matching technique is developed, i.e., the solutions of the Helmholtz wave equations in the ferrite sphere and in the surrounding areas are deduced in the form of infinite summation of spherical, cylindrical, and general Cartesian modes, respectively. The point-matching method is then used on the interface between the ferrite sphere and air within the cylindrical junction, as well as the interface between the junction and waveguides to numerically obtain the coefficients of different orders of basis functions of the field. Therefore, the field distributions, as well as the characteristics of the circulator, are numerically calculated and good agreement is observed between the numerical results and measured data.

Reconstruction of roughness profile of fractal surface from scattering measurement at grazing incidence

As a scalar wave is incident upon a rough surface at grazing angle, the Helmholtz wave equation can be approximated by the parabolic equation. Inverse scattering has been formulated in the form of a pair of coupled integral equations in two unknown functions, i.e., surface current and surface roughness. The aim of this article is to implement this approach for profile reconstruction of complex fractal rough surface. The fractal rough surfaces described by a band-limited Weieztrass–Mandelbrot function are numerically realized using the Monte Carlo method. As the scattering fields along a line parallel to the mean surface are measured or solved with a scalar wave incidence at low grazing angle, the rough surface profile is progressively inverted. Numerical examples show that the reconstructed profile and its parameters such as the fractal dimension, roughness variance, and correlation length are well matched with the simulated surfaces.

Effects of irregular terrain on waves-a stochastic approach

This paper treats wave propagation over an irregular terrain using a statistical approach. Starting with the Helmholtz wave equation and the associated boundary conditions, the problem is transformed into a modified parabolic equation after taking two important steps: (1) the forward scatter approximation that transforms the problem into an initial value problem and (2) a coordinate transformation, under which the irregular boundary becomes a plane. In step (2), the terrain second derivative enters into the modified parabolic equation. By its location in the equation, the terrain second derivative can be viewed as playing the role of a fluctuating refractive index, which is responsible for focusing and defocusing the energies. Using the propagator approach, the solution to the modified parabolic equation is expressed as a Feynman's (1965) path integral. The analytic expressions for the first two moments of the propagator have been derived. The expected energy density from a Gaussian aperture antenna has been analytically obtained. These expressions show the interplay of three physical phenomena: scattering from the irregular terrain; Fresnel phase interference; and radiation from an aperture. In particular the obtained expected energy density expression shows contributions from three sources: self energy of the direct ray; self energy of the reflected ray; and cross energy arising from their interference. When sufficiently strong scattering from random terrain may decorrelate the reflected ray from the direct ray so that the total energy density becomes the algebraic sum of two self energies. Formulas have been obtained to show the behavior of the energy density function.

Regular Structures in Self-Gravitating Dusty Plasmas

The formation of nonlinear structures in a self-gravitating dusty plasma is considered when the dust dynamics includes the electrostatic and gravitational forces on an equal footing. Here, we obtain a Helmholtz wave equation, which describes the propagation of stationary gravitational-electrostatic waves. The characteristic lengths of the space regularities, as found here, are much smaller than the wavelengths of the gravitational-electrostatic waves. Furthermore, it is shown that consideration of a uniform dusty plasma rotation around a fixed axis, gives rise to a circular rotating filament structure. Finally, the problem of the Jeans instability for circular waves in a rotating dusty plasma is generalized, and it is found that the purely growing Jeans instability assumes the form of a periodic instability.

Solution of Three-Dimensional Helmholtz Equation By Multipole Theory Method

A new approach, the multipole theory (MT) method, is presented for computation of three-dimensional (3-D) Helmholtz wave equations. By the mathematical deduction, the generalized MT formula and its applied laws are derived. In order to check the accuracy of this method, the new approach has been applied to computation of two resonant cavities with analytical solutions. Furthermore, the MT method is applied to analysis of the resonant frequency of the reentrant cylindrical cavity which is widely employed in the design of microwave oscillators. Excellent agreement with results obtained by experiment is demonstrated. It has been shown that the MT method is an effective approach for computation of 3-D Helmholtz wave equations.

Hybrid model of scattering from eccentrically nested dielectric cylinders

This paper presents an analysis of electromagnetic field scattering from a composite consisting of N eccentric dielectric cylinders. The scattering analysis is based on the finite element method (FEM) using triangular elements for inside the composite cylinder and the boundary element method (BEM) for the unbounded region outside the composite cylinder. The fields outside the eccentric dielectric cylinders are represented in terms of the boundary integral equation. The solution of Helmholtz wave equation inside the eccentric dielectric cylinders (inhomogeneous region) is provided using the finite element method. By applying the boundary conditions between free space and the outermost layer of the eccentric dielectric cylinders, the scattering coefficients of the scattered fields are obtained. The method is very efficient since it maintains the sparsity of the resulting matrix equation. The radar cross section (RCS) of the various geometries is presented. The available numerical results in the literature are used to validate the accuracy of this method.

Scattering of Electromagnetic Waves By a Chiral Object of Ellipsoidal Shape

The potential applications of electromagnetic (EM) chirality, together with the fact that a general ellipsoid shape can simulate a variety of common objects, such as discs, needles etc., constitute the motivation of the present work. Scattering of EM waves incident on an ellipsoidal object of chiral property is analyzed using Fourier analysis, in conjunction with an integral equation technique. The inner field of the scatterer is described in terms of superposition of plane waves satisfying the vector Helmholtz wave equation inside the chiral media and a Galerkin procedure is employed to solve a volume integral equation obtained by applying the Green's theorem. A highly stable numerical code is obtained, which is verified by comparison with previously published solutions. Numerical computations have been carried out for various scatterers and interesting scattering properties of waves from chiral media are observed. Especially for a linearly polarized plane wave incidence, the subsequent change of its polarization is investigated and discussed.

New Taylor-expansion method for solving a general class of wave equations

We derive a general method for solving second-order wave equations on the basis of a fourth-order Taylor expansion, with respect to time, of the field and its time derivative. The method requires explicit evaluation of space derivatives up to fourth order of the field and its time derivative, and the time step is subject to a stability criterion, similar to the one routinely applied in conjunction with the time-domain–finite-difference method. The Taylor-series method is generalizable to vector wave equations, to wave equations involving dissipation, and to the Helmholtz wave equation. The method is equally applicable to situations involving large or small variations in the refractive index. Sample numerical solutions in one and two space dimensions are described.

Corrections to the paraxial approximation of an arbitrary free-propagation beam

A relatively simple transform from an arbitrary solution of the paraxial wave equation to the corresponding exact solution of the Helmholtz wave equation is derived in the condition that the evanescent waves are ignored and is used to study the corrections to the paraxial approximation of an arbitrary free-propagation beam. Specifically, the general lowest-order correction field is given in a very simple form and is proved to be exactly consistent with the perturbation method developed by Lax [Phys. Rev. A11, 1365 (1975)]. Some special examples, such as the lowest-order correction to the paraxial approximation of a fundamental Gaussian beam whose waist plane has a parallel shift from the z=0 plane, are presented.

Mode functions for the wide-angle approximation to the parabolic equation

The parabolic approximation to the wave equation is examined within the context of normal mode theory. In a layered waveguide, the horizontal propagation constants and modal amplitudes of the field Ψ satisfying the standard parabolic equation (SPE) approximation can be mapped exactly into the amplitudes and wave numbers of the normal modes for the field p satisfying the Helmholtz wave equation. However, this is not the case for certain other parabolic approximations, such as the wide-angle parabolic equation (WAPE) approximation. Approximate mode functions for the WAPE approximation are developed. These mode functions are then used to decompose range-independent sound-pressure fields computed using the WAPE approximation. The resulting modal coefficients and eigenfunctions obtained using the WAPE mode functions are compared with those obtained using standard normal mode theory.

Te Scattering By a Cylindrical Dielectric Obstacle Buried in a Half-Space: a H-Field-Based Solution Method

Scattering of a transverse electric (TE) wavefield by an inhomogeneous lossy dielectric cylindrical obstacle embedded in a lossy dielectric half-space is investigated. The cylinder is of arbitrary cross-section, and its axis lies parallel to the interface. The formulation involves a domain integral equation of the field obtained by applying the Green's theorem to the appropriate Helmholtz wave equations. The convolution-correlation structure of the formulation is such that the discretized counterpart of the integral equation (provided by the application of a Method of Moments where the obstacle is divided into triangular patches and where the contrast and the H-field are expanded using two different basis functions) can be efficiently handled using a FFT-based conjugate gradient solver. The validity of the approach is illustrated by comparison with results found in the literature.

Regularization of Inverse Problem for Wave Scattering Based on Sub-Band Decomposition by Wavelets.

In general, inverse problems related to scientific measurements are ill-posed due to lack of sufficient amount of measured data or due to measurement errors, and an inverse solution is commonly obtained by the minimization procedure for a pertinent functional with various regularization techniques. In this study, a new regularization method based on the sub-band decomposition by orthogonal discrete wavelet transform is proposed for inverse scattering problem. The data representing the distribution of the unknown parameters are decomposed by orthogonal wavelets into components of different spatial scale, and regularization parameters are chosen separately for different components. When some a priori information regarding size and location of the scatterer are available, erroneous components with smaller scale can be damped with higher penalty number while retaining essential information of the scatterers. To demonstrate the validity of the proposed method, the results of the present inverse analysis with wavelet decomposition are compared to those obtained with ordinary inverse analysis for an inverse scattering problem associated with the two-dimensional scalar Helmholtz wave equation linearized by Born approximation.

3D Helmholtz wave equation by fictitious domain method

3D Helmholtz wave equation by fictitious domain method

Vectorial wave-matching mode analysis of integrated optical waveguides

For mode fields of integrated optical waveguides with piecewise constant and rectangular permittivity profile, Maxwell's equations reduce to the two-dimensional Helmholtz wave equation, supplemented by continuity requirements on the boundaries between different media. Two basic components of the mode field are expanded into factorizing harmonic or exponential functions, separately on each rectangular region. We determine approximations for guided modes and propagation constants by means of a minimization procedure based on a least squares expression for the mismatch in the continuity requirements. Results for the hybrid mode fields of typical sample waveguides classify this approach as competitive with and superior to alternative methods for vectorial mode analysis, although some open questions remain.