Hertz Vector Research Articles

Scattering of a plane wave by a radially stratified tilted cylinder

We calculate the intensity of the light scattered by an infinite, radially stratified cylinder. The incident light is a plane wave having an arbitrary angle of incidence and an arbitrary polarization. The Hertz potentials of the scattered wave are represented as superpositions of conical waves, and the boundary-value method is used to derive an infinite set of systems of linear equations for the expansion coefficients. The intensity and the polarization of the far-field scattered wave is expressed in terms of these expansion coefficients. Numerical results showing the angular distribution of the scattered intensity corresponding to different angles of incidence are also presented for the case of a doubly clad image-transmitting fiber illuminated by a He–Ne laser.

Hertz and Debye potentials and electromagnetic fields in general relativity

In this paper the authors extend earlier work on the application of curved space Hertz and Debye bivector potentials to the solution of Maxwell's equations for source-free electromagnetic test fields in general relativity. They classify all Hertzian schemes by considering bivector potentials which are eigenvectors of the Hodge duality operator. This approach has the advantage of simplifying the task of solving the system of coupled equations which determine the Hertz potential. They then apply their results to several Debye schemes and present a new scheme which can be used in Petrov type-D spacetimes. For each of these schemes, the one-component wave equation for the potential is given with respect to the null tetrad/Newman-Penrose formalism and, for the first time, with respect to an orthonormal basis.

Consolidated approach to two-body electromagnetic scattering.

In this paper we present a new consolidated approach to the problem of determining the electromagnetic field resulting from a finite electric current source outside two electrically different media. This approach rests on properties of generalized functions and the solution of a given problem depends essentially only on the gradients of the characteristic functions of the various regions. In particular we study the scattered fields from a flat earth, from a two-layered earth, from an earth consisting of two different adjacent electrical media (the ``mixed-path'' problem), and from a rough surface. We show that each of these problems can be modeled and solved from one set of equations; in each case we carry the analysis to a stage where it agrees with that presented by the authors in specific detailed papers that treat each problem separately. As such the solutions can be shown to agree with classical approaches. The technique used here contains a minimum of sophisticated analysis and has the following properties: The source field is arbitrary, the resulting equations directly relate the unknown electric and known source field intensities without the use of intermediate Hertz potentials, and the boundary conditions evolve directly from the formulation as auxiliary equations.

Dynamic Analysis of Microstrip Lines and Finlines on Uniaxial Anisotropic Substrates

Dyadic Green's functions in the Fourier transform spectral domain are obtained for open microstrip lines and bilateral finlines on uniaxial anisotropic substrates. These functions are written in an impedance matrix form by expressing the electric and magnetic fields in terms of Hertz vector potentials oriented along the optical axis. In combination with Galerkin's method, they are used to obtain the propagation characteristics of single and parallel coupled microstrip lines on uniaxial anisotropic substrates having the optical axis in an arbitrary direction in a transverse plane and of bilateral finlines with the three optical axis orientations of the uniaxial anisotropic substrate that result in the diagonal permittivity tensor.

The fields of a traveling wave line source in the vicinity of a half-plane

Exact expressions for the Hertz potential and electromagnetic fields radiated by an electric traveling wave line source, of arbitrary but uniform dipole orientation, in the presence of a perfectly conducting half-plane are presented. A brief discussion of the numerical evaluation of the resulting expressions is given. It is also shown that the results for the total fields can be simply modified to produce the scattered fields.

A note on the Hertz potentials in electromagnetism

A note on the Hertz potentials in electromagnetism

On the Numerical Solution of Fields in Cavities Using the Magnetic Hertz Vector

The possibility of using a representation of the electromagnetic fields in terms of a two-component magnetic Hertz vector to calculate fields and frequencies in cavities is examined. The governing equation is the vector Helmholtz equation, and the model is tested by calculating fields in axially symmetric cavities.

Reduction of dyadic Green's functions to scalar Hertz potentials for gyrotropic media

Electromagnetic fields in anisotropic media can be represented in terms of dyadic Green's functions. The four dyadic Green's functions (Green's tensors) are solutions of systems of coupled tensor equations of first order which can be transformed to second‐order tensor equations for each of the Green's tensors. For the special case of gyrotropic media (anisotropic media with a distinguished axis and electromagnetic parameters having a rotational invariance with respect to this axis, like a cold ionospheric electron plasma under the influence of earth's magnetic field) it is shown how the Green's tensors can be represented in terms of scalar Hertz potentials. So the problem of solving tensor equations can be reduced to the solution of a system of two coupled partial differential equations of second order for only two scalar potentials.

Pseudodecoupling ansatz for electromagnetic aperture coupling in three dimensions

Hertzian potentials are commonly used to describe the scattering of electromagnetic waves in three dimensions. However, when apertures are present in the scattering object, mixed boundary conditions arise and the TE and TM modes of the scattered field become coupled. Numerous attempts to decouple these TE and TM modes have led to solutions of Maxwell's equations which fail to satisfy Meixner's edge conditions. It has been found that by introducing additional gauge potentials which are homogeneous solutions of the boundary condition equations, a pseudodecoupling of the TE and TM modes can be achieved while still satisfying Meixner's edge conditions.

A note on Hertz vectors

The purpose of this note is to clarify the role of the Hertz vector in the description of a source free electromagnetic field in vacuo. In particular we show that the total energy and the total helicity of both the electric field and the magnetic field can be expressed as time derivatives of the corresponding quantities for the Hertz vector.

100kHz電磁波エネルギーの流れに及ぼす山岳の影響 : 三角形山岳モデルによる

Hertz vector values were given to the nodes upon the contour as a restraint one. Then the Hertz potential of each node in the given area was calculated using the Finite Element Method (FEM) on the flat earth far from the transmitting antenna. Those well consisted with K.A. Norton's calculated values. Next, the electro-magnetic field at the vicinity of a triangular mountain model set on the flat earth, was obtained by the FEM as well. A group of curves showing the direction of the maximum electric field vector and another group of curves indicating energy flows of the electro-magnetic wave were pictured from the result. The conclusion is such that the range suffered influence due to the mountain model which height and inclination are 250(m) and 45 degrees respectively, for example, is approximately twice as much as the height.

Scalarisation of maxwell's equations in general inhomogeneous bianisotropic media

Electromagnetic field representations in spatially inhomogeneous bianisotropic media are investigated. It is shown that the solution of Maxwell's equations can be reduced to the solution of a set of partial differential equations for four scalar potentials from which the fields can be derived. To obtain an analogous representation as in isotropic media — where two scalar Hertz potentials are sufficient to determine the electromagnetic field — one has to impose restrictions on the constitutive electromagnetic parameters of these inhomogeneous and bianisotropic media. The applicability of the scalar Hertz-potential method is limited to bigyrotropic media which are inhomogeneous only in the direction of the bigyrotropic media's distinguished axis. Finally, inhomogeneous gyrotropic media and moving isotropic media serve as examples for the outlined method.

Proca and electromagnetic fields

In the framework of the proper orthochronous Lorentz group, the old connection is revived between the electromagnetic field characterized by a self-dual tensor and a traceless second-rank spinor obeying the Proca equation. The relationship between this spinor and the Hertz potential also considered as a self-dual tensor is emphasized. The extension of this formalism to meet the covariance under the full Lorentz group is also discussed.

Equivalence of self-consistency and conservation of energy in the scattering of an electromagnetic wave by a plane of dipoles

The scattering of a harmonic electromagnetic plane wave by a plane of induced dipoles has been rigorously treated by means of the Hertz vector. We have shown that only a self-consistent solution, which takes into account cooperative effects among scatterers, is able to conserve energy.

Conserved quantities in electromagnetic theory

A method of constructing conserved quantities associated with an electromagnetic field is given and the particular ones which can arise from field intensities, vector potentials and Hertz potentials are obtained and related to known results.

A study of electromagnetic field around delay line having various constructions of outer conductor

AbstractWith a view to application of the electromagnetic field around a delay line to the mobile communication, a method of calculation and an analysis of characteristics of such a field around a delay line with various configurations of outer conductors have been presented. For the calculation of the electromagnetic field around the delay line, the magnetic and electric Hertz vectors are used. Variations of the propagation constant and the electromagnetic field of the line with and without the shielding effect of the outer conductor have been calculated. The effect on the magnetic and electric sources on the external surface of the line has been studied. The electromagnetic field around the delay line can be derived as the sum of the Hertz vectors generated by various sections of the line. It can be characterized as the convergent field when the line is uniform and homogeneous. In this paper, a new model of the line for the radiation field has been proposed in which shielded and unshielded delay lines are placed periodically. Both models are compared for the effect of the phase constant and the end effect. The effects of the nonuniformity in the line and also the reflection at the end and the curved section on the convergent field as well as the attenuation constant are studied in detail and verified experimentally. Finally, it is pointed out that the inductive field is predominant in the case of an outer conductor with an imperfect shielding effect.

Enhanced fields on rough surfaces: dipolar interactions among particles of sizes exceeding the Rayleigh limit

Amplified electromagnetic fields generated by a surface of finitely sized metal or dielectric particles are calculated. Regular arrays of particles produced by lithographic techniques and stochastic particle distributions that occur, e.g., in island films, are discussed. Retarded dipolar interactions between the particles are explicitly taken into account. Particles of finite size are considered for which dynamic depolarization and radiation damping effects are important. Limits of validity of the present approach are indicated. The total electromagnetic field from the surface is calculated by superposition: Scalar potentials characterizing a single particle are convoluted with a distribution function describing the particle positions. The surface Hertz vector is obtained from the single-particle Hertz vector by convolution with a two-dimensional Shah function representing the array or with the autocorrelation function of the stochastic surface. A plane-wave description of the dipolar fields is used, whereby the convolution is transformed into a simple multiplication in Fourier space. Cylindrical, general spheroidal, and spherical shapes are considered for the individual particle. Particle dipole moments are obtained by a self-consistent procedure. Dipolar interactions result in shifts and broadening of the particle plasmon resonances, which are responsible for the local intensity enhancement. A set of universal curves is given from which shift and broadening can be calculated for particles of all sizes and shapes. Extrema in the dipolar interactions arise when grating orders change from radiative to evanescent character. The strong variation of the Raman enhancement with angle and wavelength in the vicinity of these extrema is clearly predicted from the Hertz-vector calculation. The formalism described permits one to calculate electromagnetic properties of the surface and enhancement factors for any electromagnetic process occurring at or near the surface. As examples, the calculation of reflectivities for s-and p-polarized excitation and surface-enhanced Raman-scattering cross sections are discussed.

Separability of the Dirac equation in a class of perfect fluid space-times with local rotational symmetry

Chandrasekhar’s technique for separation of the Dirac equation in the Kerr background is applied to perfect fluid space-times with local rotational symmetry. These space-times fall into three distinct types. It is found that in case (1) the Dirac equation separates if the space-time is at least ‘‘locally static’’ while in case (3) it separates if the space-time is at least ‘‘locally diagonal,’’ in contrast to the massless case where Dhurandhar, Vishveshwara, and Cohen showed that the Hertz potential is separable in all cases. In case (2), however, the Dirac equation is separable in all those cases where the Hertz potential for neutrinos is separable.

The effect of a dielectric cover on the current distribution and input impedance of printed dipoles

The effect of the thickness and relative permittivily of a dielectric cover on a printed microstrip dipole has been analyzed. It is shown that the current distribution and the input impedance are, in general, very sensitive to variations of the cover parameters. For a dielectric plate with a constant thickness the dipole resonant length decreases substantially with an increase of the relative permittivity. Because of the limited bandwidth presented by single-element microstrip antennas the effects of the dielectric cover on the design of these antennas have to be carefully considered. For the calculation of the current distribution, the Hertz vector potential associated with the problem was determined for an element of current located in a stratified medium with four layers. Pocklington's integral equation was solved for the currents, using Galerkin's method with piecewise-sinusoidal expansion and weighting functions.

Null strings and Hertz potentials

A covariant expression for the massless spinor fields in terms of Hertz potentials is given, assuming that the space-time admits a congruence of null strings defined by a multiple Debever–Penrose spinor. The metric of these spaces is also given in a covariant form.