Hertz Vector Research Articles

Separable electromagnetic perturbations of rotating black holes

We identify a set of Hertz potentials for solutions to the vector wave equation on black hole spacetimes. The Hertz potentials yield Lorenz gauge electromagnetic vector potentials that represent physical solutions to the Maxwell equations, satisfy the Teukolsky equation, and are related to the Maxwell scalars by straightforward and separable inversion relations. Our construction, based on the GHP formalism, avoids the need for a mode ansatz and leads to potentials that represent both static and non-static solutions. As an explicit example, we specialise the procedure to mode-decomposed perturbations of Kerr spacetime and in the process make connections with previous results.

Second-order perturbations of Kerr black holes: Formalism and reconstruction of the first-order metric

Motivated by gravitational wave observations of binary black hole mergers, we present a procedure to compute the leading-order nonlinear gravitational wave interactions around a Kerr black hole. We describe the formalism used to derive the equations for second-order perturbations. We develop a procedure that allows us to reconstruct the first-order metric perturbation solely from knowledge of the solution to the first-order Teukolsky equation, without the need of Hertz potentials. Finally, we illustrate this metric reconstruction procedure in the asymptotic limit for the first-order quasinormal modes of Kerr. In a companion paper [J. L. Ripley et al., Phys. Rev. D 103, 104018 (2021)] we present a numerical implementation of these ideas.

Asymptotic solution for the electromagnetic scattering of a vertical dipole over plasmonic and non‐plasmonic half‐spaces

A new asymptotic solution for the scattered electromagnetic fields of a vertical Hertzian dipole antenna in the presence of an imperfectly conducting half-plane for ordinary and plasmonic media is proposed. The scattered electric and magnetic field components are calculated from the intermediate Hertz potential expressed in terms of the Fourier-Bessel transforms associated with the Sommerfeld-type integral, which is difficult to evaluate due to the singularities of the integrand near the integration path and its oscillatory and slowly decaying integrand. Using the modified saddle point method, an approximate closed-form solution of the far-zone scattered electromagnetic fields including surface waves is presented. The new formulations are applied to calculate radiation patterns of different impedance half-planes for both ordinary and plasmonic media, that is, seawater, silty clay soil, silty loam soil and lake water as ordinary, and silver and gold as plasmonic media. Furthermore, a numerical evaluation of the proposed solution at various frequencies and comparisons with two alternative state of the art solutions shows that the proposed solution has higher accuracy in terms of the normalised root-mean-square error and the normalised maximum absolute error for plasmonic and non-plasmonic structures.

Recurrent relationships for reflection coefficients of electromagnetic wave potential representation modes in manylayered spherical environment excitation problem

In theoretical assessments of the electrodynamic characteristics of multilayer screens of electromagnetic waves, the concept of the input impedance of a wave on the surface of the screen is used [1]. In [2], an approach was proposed that is not based on this characteristic, and direct recurrence relations were obtained both for the reflection coefficient of a plane wave from a multilayer screen and for the coefficient of transmission of such a wave through this screen. In [3], this approach was applied to solving problems of excitation of a layered medium by localized sources. In this paper, we generalize the approach to the case of a multilayer medium with another simple layer geometry (with interface in the form of concentric spherical shells). The solution of excitation problems is based on the representation of the Hertz potentials of waves of electric (magnetic) types in the form of series in spherical harmonics. Each harmonic defines the angular distribution of radially converging and diverging modes of the representation of potentials in layers free from external sources. The recurrence relations for the reflection coefficients of convergent (diverging) modes of representation when sequentially adding interfaces are a consequence of the boundary conditions. One of the equivalent forms of these relations is derived from the condition for the existence of a nontrivial solution of a system of linear equations for the amplitudes of converging (diverging) modes in adjacent layers. The other is based on a simple physical model of the process of interaction of a converging (diverging) mode in a layer with a layered structure, which it covers (which covers it). For a field source inside one of the layers, one can determine the representation of potentials that this source would excite in an infinite medium with permeabilities similar to a layer. The amplitudes of the modes of representation of this «primary» field and the reflection coefficients of the modes of representation from structures at the internal and external boundaries of the layer, found from recurrence relations, determine the amplitudes of the modes of the total field of the layer incident on these boundaries. To determine the amplitudes of the modes of representation of the potentials of the fields in the remaining layers, recurrence relations can also be used, in a certain sense inverse to the recurrence relations for reflection coefficients. If a layered structure is excited by a plane wave (scattering problem), the total field potentials (of the electric and magnetic types) are the sum of radially converging modes in the potentials describing the incident plane wave and radially diverging modes formed by the reflection of these converging mod from structure. With the proposed approach, the complexity of solving the excitation problems of the considered layered structure remains almost unchanged with an increase in the number of layers. In essence, only the number of repetition cycles of the obtained recurrence relations increases. The proposed approach is conceptually simple, the final recurrence relations are physically transparent. The work does not represent the full facilities of the approach. Within the framework of this approach, recurrence relations for the permeability coefficients of multilayer spherical shells of the structure under consideration, recurrence relations during its layer-by-layer formation or formation from several multilayer shells can be found.

An Analytical Solution of the Electric Field Excited by a Vertical Electric Dipole Above a Lossy Half-Space: From Radio to Microwave Frequencies

In this article, an analytical solution of the electromagnetic radiation from a vertical electric dipole (VED) above a lossy half-space is proposed. The expression for the traditional solution is associated with the Sommerfeld-type integral, which is difficult to evaluate due to its highly oscillating and slowly decaying integrand. The solution for the intermediate Hertz potential is decomposed into two integrals and a rigorous closed-form solution in the near and far-field regions are presented for each term. Then, the scattered electric-field components are calculated from the intermediate Hertz potential. A numerical evaluation of the solution for different lossy half-spaces, such as seawater, wet earth, dry earth, and lake water, validates the accuracy of the proposed solution at various frequencies, from radio to microwave frequencies, as well as different distances from the antenna in the near and far-field regions.

A novel coplanar system with a minimal anisotropic effect

To obtain the response of the novel coplanar system, we use an analytic method to solve horizontal component of electromagnetic field in the anisotropic stratigraphic model by introducing the magnetic Hertz vector and the magnetic Hertz scalar from Maxwell equations. This paper examines anisotropic response characteristics between novel coplanar system and traditional coplanar system. The negative response area of the proposed coplanar system is decreased. Compared this with traditional coplanar system, the anisotropic effects have the same trend; the affected amplitude of novel coplanar system is far less than that of the traditional coplanar system. The results of experiments demonstrate the superiority of the proposed coplanar system.

Design of One-Eighth Spherical Dielectric Resonator Antenna for 5G Applications

A one-eighth spherical surface dielectric resonator antenna (OESS-DRA) is proposed first time. To analyze this thin shell structure, electromagnetic fields distribution is discussed by using the Hertz vector and boundary conditions. We simplify the solution of Maxwell equations by using Hertz vector because the polarization current is primary in the OESS-DRA. The Hertz vector can be expressed the current better. This proposed antenna is composed of one-eighth spherical dielectric shell fed by coaxial probe; and the ground with metal-corner-reflector makes the antenna with high gain and directional radiation properties. To verify this design, a fabricated sample is tested; results show the antenna covering bandwidth 4.56-6.88GHz, a high gain greater than 9.5dBi (peak gain 11.8dBi at 5.0GHz), and a high radiation efficiency over 85% in the entire working frequency band, as well. The OESS-DRA can be a good candidate in 5 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> generation communication applications.

A Computational Channel Model for Magnetic Induction‐Based Subsurface Applications

AbstractThere are many underground applications based on magnetic fields generated by an oscillating magnetic source. For them, a magnetic dipole in a three‐layered region with upper semi‐infinite air layer can be a convenient idealization used for their planning, development, and operation. Solutions are in the form of the well‐known Sommerfeld integral expressions that can be evaluated by numerical methods. A set of field expressions to be numerically evaluated by an efficient algorithm are not collected comprehensively yet, or at least in a directly usable form. In this paper, the explicit magnetic field solutions for the vertical magnetic dipole and the horizontal magnetic dipole for a general source‐observer location are derived from the Hertz vector. They can be properly combined to model the problem of a tilted magnetic dipole source for horizontally or inclined stratified media. As a result, a complete set of integral equations of the Sommerfeld type valid from the near zone to the far zone are formulated. A method for numerical evaluation of the field expressions for high accurate computations is described. The numerical results are validated using the finite element method for all the possible source‐receiver configurations and three well‐spanned frequencies of typical subsurface applications. Both numerical solutions agree according to the normalized root‐mean‐square error‐based fit metric. Numerical results for two cases of study are presented to see its usefulness for subsurface applications. A MATLAB implementation of the mathematical description outlined in this paper and the proposed evaluation method is freely available for download.

Electromagnetic fields on Kerr spacetime, Hertz potentials, and Lorenz gauge

We review two procedures for constructing the vector potential of the electromagnetic field on Kerr spacetime, namely, the classic method of Cohen & Kegeles, yielding $A^\mu$ in a radiation gauge, and the newer method of Frolov et al., yielding $A^\mu$ in Lorenz gauge. We demonstrate that the vector potentials are related by straightforward gauge transformations, which we give in closed form. We obtain a new result for a separable Hertz potential $H^{\mu \nu}$ such that $A_{\text{lor}}^\mu = \nabla_\nu H^{\mu \nu}$.

Symmetries of linearized gravity from adjoint operators

Based on operator identities and their formal adjoints, we derive two symmetry operators for the linearized Einstein operator on vacuum backgrounds of Petrov type D and, in particular, the Kerr spacetime. One of them is of differential order four and coincides with a result of Cohen and Kegeles. The other one is a new operator of differential order six. The corresponding operator identities are based on the Teukolsky equation and the Teukolsky-Starobinsky identities, respectively. The method applies to other field equations as well, which is illustrated with the Maxwell equation. The resulting symmetry operators are connected to Hertz and Debye potentials as well as to the separability of the Teukolsky equation for both Maxwell and linearized gravity.

Radiation of a charge moving along a corrugated surface with a relatively small period

We consider electromagnetic radiation of a charged particle bunch moving uniformly along a corrugated planar metallic surface. It is assumed that the wavelengths under consideration are much larger than the period and the depth of corrugation. Using the method of the equivalent boundary conditions we obtain the Fourier-transform of the Hertz vector. It is demonstrated that the ultra-relativistic bunch excites the surface waves, whereas the volume radiation is absent. Fourier-transforms of the surface wave components and spectral density of energy losses are obtained and analyzed.

Derivation and application of a Green function propagator suitable for nonparaxial propagation over a two-dimensional domain.

The goal of optical simulation is to determine the performance characteristics of an optical system from a knowledge of its physical construction and how it affects light sent through it. To produce meaningful results efficiently, two simulation approaches are available for passing light through a system, geometrical raytracing and wave optics. Within the wave optics realm, there are many techniques for determining the optical fields within a system, both numerical and analytical. A few of the numerical techniques are finite-difference, finite-element, and FFT-based; analytical techniques include modal expansions, coupled wave theory, series expansions, and Green function propagators. A propagator is a function that gives the light fields at any specified location if they are known at a source location; this is possible because the light fields, electric and magnetic, satisfy a differential equation, in the case of time harmonic fields, the Helmholtz equation. The propagator is a transfer function for the fields and often takes the form of an integral, in which case, the integrand is a product of the transfer function with the source field distribution, and the integration is performed over the source field coordinates. The integrand transfer function, also known as a Green function or propagation kernel, is a solution of the Helmholtz equation. An approximation is often used in finding a solution to the Helmholtz equation, called the paraxial approximation, in which the second derivative in the propagation direction is dropped. If no approximation is made, and all second derivatives are kept, the solution is nonparaxial. In the present paper, a Green function for the propagator of the Helmholtz equation over two-dimensional domains is derived, differing in functional form from previous work on two-dimensional propagation. An angular spectrum integral is evaluated and the resulting Green function, the propagator kernel, is a nonparaxial analytic solution of the Helmholtz equation. The propagator could be applied directly to the electric and magnetic field components; instead, it is applied to the Hertz vector components. The Hertz vector is a potential function, similar to the vector potential, defined such that the electric and magnetic fields are found by taking derivatives of it. An advantage of the Hertz vector is that only it needs be propagated, versus two, electric and magnetic, vectors. In this paper, the derived propagator is applied to Hertz vector components defined by Legendre polynomial expansions, and derivatives are taken of the propagated Hertz vector components to calculate the associated electric and magnetic fields. The Green function propagator and all field quantities produced by its application are closed form analytic expressions.

Canonical energy and Hertz potentials for perturbations of Schwarzschild spacetime

Canonical energy is a valuable tool for analyzing the linear stability of black hole spacetimes; positivity of canonical energy for all perturbations implies mode stability, whereas the failure of positivity for any perturbation implies instability. Nevertheless, even in the case of 4D Schwarzschild spacetime—which is known to be stable—manifest positivity of the canonical energy is difficult to establish, due to the presence of constraints on the initial data as well as the gauge dependence of the canonical energy integrand. Consideration of perturbations generated by a Hertz potential would appear to be a promising way to improve this situation, since the constraints and gauge dependence are eliminated when the canonical energy is expressed in terms of the Hertz potential. We prove that the canonical energy of a metric perturbation of Schwarzschild that is generated by a Hertz potential is positive. We relate the energy quantity arising in the linear stability proof of Dafermos, Holzegel and Rodnianski (DHR) to the canonical energy of an associated metric perturbation generated by a Hertz potential. We also relate the Regge–Wheeler variable of DHR to the ordinary Regge–Wheeler and twist potential variables of the associated perturbation. Since the Hertz potential formalism can be generalized to a Kerr black hole, our results may be useful for the analysis of the linear stability of Kerr.

Elegant Laguerre–Gaussian beams—formulation of exact vector solution

In photonic applications of optical beams, their transverse cross-section should be often narrow, with a diameter in their waist of the order of one wavelength or even less. Within this range, the paraxial approximation of beam fields is not valid and standard corrections by field expansions with respect to a small parameter are not efficient as well. Thus, still there is a need for more accurate beam field description. In this report, an exact vector solution for free-space propagation is given in terms of elegant Laguerre–Gaussian beams. The analysis starts from the known paraxial field approximation and next, through bidirectional field transformation and application of a Hertz potential leads to an exact vector solution. The role of the paraxial solution in construction of the exact solution is elucidated. The method works well not only in cases of free-space propagation but also in description of beam interactions with planar interfaces and multilayers.

An improved structure of coplanar coil system and its logging response

An improved structure of coplanar system with a narrow negative response, minimal skin effect and horns is presented, which consist of two transmitters and one receiver coil. To obtain the response of the improved coplanar system, we use an analytic method to solve Hertz potentials by applying tangential continuity in layer boundary and get the horizontal component of magnetic field intensity. Further, the models for received signals are analysed in the cylindrically and the horizontally layered formation respectively. Compared with the traditional coplanar configuration, the negative response area of the new coplanar system is significantly decreased, the responses are approximately linear and the spatial resolution is enhanced by two times. Furthermore, the improved coplanar system mechanisms are revealed for reducing skin effect and negative response through analysis of the distribution characteristics of eddy current of the transmitter coil. The results of experiments demonstrate the superiority of the proposed coplanar system.

Optical helicity and Hertz vectors

We study the conserved quantity associated with the dual symmetry of the Maxwell equations, called the optical helicity, by means of transverse Hertz vectors. In the presence of charges, its evolution yields the integral of E⋅B which is the anomaly term for chiral fermions. We also discuss the helicity change in condensed matter systems where topological magnetoelectric effect emerges. An alternative expression of the optical helicity is also found. Lastly, a dual symmetric Hertz Lagrangian is constructed and its conserved charge is derived.

The phase diagram and melting scenarios of two-dimensional Hertzian spheres

ABSTRACTWe present computer simulations of a system of purely repulsive soft colloidal particles interacting via the Hertz potential and constrained to a two-dimensional plane. This potential describes the elastic interaction of weakly deformable bodies and can be a reliable model for a qualitative description of behaviour of soft macromolecules, such as globular micelles and star polymers. We have found a large number of ordered phases, including dodecagonal quasicrystal, and analysed the melting scenarios of low-density triangle and square phases. It is interesting that depending on the position on the phase diagram, the system can melt both through the first-order transition and in accordance with the Berezinskii–Kosterlitz–Thouless–Halperin–Nelson–Young scenario (two continuous transitions with an intermediate hexatic phase) as well as according to recently proposed two-stage melting with the first-order hexatic–isotropic liquid transition and continuous solid–hexatic transition. We have also demonstrated the possibility of a tricritical point on the melting line and have found the line of a water-like density anomaly on the phase diagram.

Scattering by multiple cylinders located on both sides of an interface

The solution for scattering by multiple parallel infinite cylinders located in adjacent half spaces with dissimilar refractive index is presented in this paper. The incident radiation is an arbitrarily polarized plane wave propagating in the upper half space in the plane perpendicular to the axis of the cylinders. The formulation of the electromagnetic field vectors utilized Hertz potentials that are expressed in terms of an expansion of cylindrical wave functions. It accounts for the near-field multiple scattering, Fresnel effect at the interface, and interaction between cylinders in both half spaces. Analytical formulas are derived for the electromagnetic field and Poynting vector in the far-field. The present solution provides the theoretical framework for deducing the solutions for scattering by cylinders located on either side of an interface irradiated by a propagating or an evanescent incident wave. Deduction of these solutions from the present formulation is demonstrated. Numerical results are presented to illustrate the frustration of total internal reflection and scattering of light beyond the critical angle by nanocylinders located in either or both half spaces.

The equilibrium phase in heterogeneous Hertzian chains

We examine the long-term behaviour of non-integrable, energy-conserved, 1D systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. We previously showed that in homogeneous configurations of such systems, energy is equipartitioned at sufficiently long times, thus these systems ultimately reach thermal equilibrium. Here we expand on our previous work to show that heterogeneous configurations of grains also reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. We investigate the transition to equilibrium in detail and introduce correlation functions that indicate the onset of the transition.

Wave propagation characteristics of a magnetic granular chain

We investigate the wave propagation characteristics of a horizontal alignment of magnetic grains under a non-uniform magnetic field. The magnetic force of each grain is obtained using Maxwell's principle. The contact interaction of grains is based on Hertz potential. The effects of magnetic field strength on the dynamic responses of a granular chain under strong, intermediate, and weak amplitudes of incident impulses in comparison with static precompression force are studied. Different wave propagation modes induced by the magnetic field are observed. The applied field strength demonstrably reinforces the granular-position-dependent behaviors of decreasing amplitude and increasing wave propagation velocity. The magnetic field-induced features of a magnetic granular chain have potential applications in adaptive structures for shock attenuation.