Point Electric Dipole Research Articles

Second Gradient Electromagnetostatics: Electric Point Charge, Electrostatic and Magnetostatic Dipoles

In this paper, we study the theory of second gradient electromagnetostatics as the static version of second gradient electrodynamics. The theory of second gradient electrodynamics is a linear generalization of higher order of classical Maxwell electrodynamics whose Lagrangian is both Lorentz and U ( 1 ) -gauge invariant. Second gradient electromagnetostatics is a gradient field theory with up to second-order derivatives of the electromagnetic field strengths in the Lagrangian. Moreover, it possesses a weak nonlocality in space and gives a regularization based on higher-order partial differential equations. From the group theoretical point of view, in second gradient electromagnetostatics the (isotropic) constitutive relations involve an invariant scalar differential operator of fourth order in addition to scalar constitutive parameters. We investigate the classical static problems of an electric point charge, and electric and magnetic dipoles in the framework of second gradient electromagnetostatics, and we show that all the electromagnetic fields (potential, field strength, interaction energy, interaction force) are singularity-free, unlike the corresponding solutions in the classical Maxwell electromagnetism and in the Bopp–Podolsky theory. The theory of second gradient electromagnetostatics delivers a singularity-free electromagnetic field theory with weak spatial nonlocality.

Lattice Resonances in Optical Metasurfaces With Gain and Loss

Periodic lattices of strongly scattering objects coupled to active media are of central importance in applied nanophotonics, serving as light-emitting metasurfaces of tailored emission properties and promising an attractive platform for testing novel physical concepts and realization of unprecedented light-shaping functions. We provide an overview of the semianalytical Green function method with Ewald lattice summation applied to the investigation of surface lattice resonances in periodic arrays of resonant nanoscatterers with gain and loss. This theory is meant as a minimal model for plasmonic lattices and metasurfaces with gain: minimal in complexity, yet sufficiently rich to be a self-consistent, fully retarded multiple scattering model. It enables to include the electromagnetic interactions between electric and/or magnetic point dipoles of arbitrary orientation and arrangement, taking into account retardation and tensorial nature of these interactions and including radiation damping. It gives access to the far-field observables (reflection/transmission), as well as to the photonic band structure of guided modes. At the same time, it does not violate the optical theorem, as opposed to the commonly used tight-binding or quasi-static models. After extending the lattice Green function formalism to include gain and loss in the unit cell, we demonstrate the effects of parity-time (PT) symmetry breaking in active-lossy plasmonic arrays: the emergence of exceptional points, nontrivial topology of photonic bands, diverging effective unit-cell polarizability, and spin polarization in the PT-broken phase.

Spatial coherence of the thermal emission of a sphere

Analytical expressions for calculating the energy density and spatial correlation function of thermal emission by a homogeneous, isothermal sphere of arbitrary size and material are presented. The spectral distribution and the power law governing the distance-dependent energy density are investigated in the near-field and far-field regimes for silicon carbide, silicon and tungsten spheres of various size parameters ranging from X = 0.002 to 5. The spatial coherence of thermal field emitted by spheres is also studied in both radial and polar directions. The energy density follows a power law of d^-2 (d is the observation distance) in the far field for all sizes and materials. The power law in the near field is strongly dependent on the material, size parameter, and the ratio d/a (a is the sphere radius). In the near field, the energy density follows a power law of d^-6 when X<<1 and d/a>>1 (similar to an electric point dipole). With increasing X or decreasing d/a, the contribution of multipoles to the energy density increases resulting in an increase in the power of d until the power law converges to that for a semi-infinite medium. The spatial correlation length in the radial direction is in the orders of $\lambda$, 0.1$\lambda$, and 0.001$\lambda$ in the far field, intermediate near field, and extreme near field, respectively. The correlation angle in the extreme near field is strongly dependent on the sphere size parameter, such that it decreases by three orders of magnitude (from 0.5$\pi$ to 0.001$\pi$) when X increases from 0.002 to 5. In the intermediate near field and far field, the correlation angle retains the same order of magnitude (0.15$\pi$ - 0.7$\pi$) for all considered Xs. While the excitation of dipolar localized surface phonons (LSPhs) does not affect the correlation length and angle, the multipolar LSPhs reduce the spatial coherence in both directions.

Three-dimensional electric potential induced by a point singularity in a multilayered dielectric medium

A simple and effective method is proposed to derive the three-dimensional electric potential induced by a point singularity of any type in an N-phase dielectric medium composed of N−2 intermediate dielectric layers of equal thickness encased in two semi-infinite dielectric media. The point singularity can include a point charge or a point electric dipole. The original boundary value problem for the N-phase medium is reduced to the determination of a single unknown three-dimensional harmonic function through satisfaction of the continuity conditions across all of the N−1 perfect planar interfaces. The single harmonic function can be completely determined after analytically solving the resulting linear recurrence relations, which are independent of the type and the specific location of the singularity. When the singularity is a point charge, we obtain the self-energy of the point charge expressed in terms of a single function and the Coulomb force on the point charge expressed in terms of the negative derivative of this function.

Point-dipole approximation for small systems of strongly coupled radiating nanorods

Systems of closely-spaced resonators can be strongly coupled by interactions mediated by scattered electromagnetic fields. In large systems the resulting response has been shown to be more sensitive to these collective interactions than to the detailed structure of individual resonators. Attempts to describe such systems have resulted in point-dipole approximations to resonators that are computationally efficient for large resonator ensembles. Here we provide a detailed study for the validity of point dipole approximations in small systems of strongly coupled plasmonic nanorods, including the cases of both super-radiant and subradiant excitations, where the characteristics of the excitation depends on the spatial separation between the nanorods. We show that over an appreciable range of rod lengths centered on 210 nm, when the relative separation kl in terms of the resonance wave number of light k satisfies klgtrsim pi /2, the point electric dipole model becomes accurate. However, when the resonators are closer, the finite-size and geometry of the resonators modifies the excitation modes, in particular the cooperative mode line shifts of the point dipole approximation begin to rapidly diverge at small separations. We also construct simplified effective models by describing a pair of nanorods as a single effective metamolecule.

Cooperative field localization and excitation eigenmodes in disordered metamaterials

We investigate numerically and experimentally the near-field response of disordered arrays comprising asymmetrically split ring resonators that exhibit strong cooperative response. Our simulations treat the unit cell split ring resonators as discrete pointlike oscillators with associated electric and magnetic point dipole radiation, while the strong cooperative radiative coupling between the different split rings is fully included at all orders. The methods allow to calculate local field and Purcell factor enhancement arising from the collective electric and magnetic excitations. We find substantially increased standard deviation of the Purcell-enhancement with disorder, making it increasingly likely to find collective excitation eigenmodes with very high Purcell factors that are also stronger for magnetic than electric excitations. We show that disorder can dramatically modify the cooperative response of the metamaterial even in the presence of strong dissipation losses as is the case for plasmonic systems. Our analysis in terms of collective eigenmodes paves a way for controlled engineering of electromagnetic device functionalities based on strongly interacting metamaterial arrays.

Electrostatic and magnetostatic fields of point dipoles revisited

After reviewing how the Dirac delta contributions to the electrostatic and magnetostatic fields of a point electric dipole and a point magnetic dipole are usually introduced, we present an alternative procedure for obtaining these terms based on a regularization prescription similar to that used in the computation of the transverse and longitudinal delta functions. We think this method may be useful for the students in other analogous calculations.

Mitigating VHF Lightning Source Retrieval Errors

AbstractThe problem of inferring the location and time of occurrence of a very high frequency (VHF) lightning source emission from Lightning Mapping Array (LMA) network time-of-arrival (TOA) measurements is closely examined in order to clarify the cause of retrieval errors and to determine how best to mitigate these errors. With regard to this inverse problem, the previous literature lacks a comprehensive discussion of the associated forward problem. Hence, the forward problem is analyzed in this study to better clarify why retrieval errors increase with increasing source horizontal range and/or decreasing source altitude. Further insight is obtained by performing carefully designed Monte Carlo inversion simulations that provide specific retrieval error plots, which in turn lead to clear recommendations for mitigating retrieval errors. Based on all of the numerical results, the following strategies are recommended for mitigating retrieval errors (when possible, and without obstructing the line of sight): expand the horizontal extent of the LMA network, maximize the vertical sensor baseline by using mountainous terrain if available, and improve TOA measurement timing accuracy. Adding sensors to the network is relatively ineffective, unless of course the addition of sensors expands the horizontal extent and/or vertical baseline of the network. It is also shown how the standard retrieval method can be generalized by considering, in addition to the regular (unpolarized) point VHF source, the polarized transient very low frequency/low frequency (VLF/LF) electric point dipole source. Multiple observations (i.e., VHF arrival time and power, and VLF/LF arrival time and electric field amplitude) are simultaneously implemented into the new generalized mathematical framework, and the potential benefits are indicated.

Model of directed lines for square ice with second-neighbor and third-neighbor interactions

The investigation of the properties of nanoconfined systems is one of the most rapidly developing scientific fields. Recently it has been established that water monolayer between two graphene sheets forms square ice. Because of the energetic disadvantage, in the structure of the square ice there are no longitudinally arranged molecules. The result is that the structure is formed by unidirectional straight-lines of hydrogen bonds only. A simple but accurate discrete model of square ice with second-neighbor and third-neighbor interactions is proposed. According to this model, the ground state includes all configurations which do not contain three neighboring unidirectional chains of hydrogen bonds. Each triplet increases the energy by the same value. This new model differs from an analogous model with long-range interactions where in the ground state all neighboring chains are antiparallel. The new model is suitable for the corresponding system of point electric (and magnetic) dipoles on the square lattice. It allows separately estimating the different contributions to the total binding energy and helps to understand the properties of infinite monolayers and finite nanostructures. Calculations of the binding energy for square ice and for point dipole system are performed using the packages TINKER and LAMMPS.

Magnetic Dipole Scattering from Metallic Nanowire for Ultrasensitive Deflection Sensing.

It is generally believed that when a single metallic nanowire is sufficiently small, it scatters like a point electric dipole. We show theoretically when a metallic nanowire is placed inside specially designed beams, the magnetic dipole contribution along with the electric dipole resonance can lead to unidirectional scattering in the far field, fulfilling Kerker's condition. Remarkably, this far-field unidirectional scattering encodes information that is highly dependent on the nanowire's deflection at a scale much smaller than the wavelength. The special roles of small but essential magnetic response along with the plasmonic resonance are highlighted for this extreme sensitivity as compared with the dielectric counterpart. In addition, the same essential role of the magnetic dipole contribution is also presented for a very small metallic nanosphere.

Point dipole and quadrupole scattering approximation to collectively responding resonator systems

We develop a theoretical formalism for collectively responding point scatterers where the radiating electromagnetic fields from each emitter are considered in the electric dipole, magnetic dipole, and electric quadrupole approximation. The contributions of the electric quadrupole moment to electromagnetically-mediated interactions between the scatterers are derived in detail for a system where each scatterer represents a linear $RLC$ circuit resonator, representing common metamaterial resonators in radiofrequency, microwave, and optical regimes. The resulting theory includes a closed set of equations for an ensemble of discrete resonators that are radiatively coupled to each other by propagating electromagnetic fields, incorporating potentially strong interactions and recurrent scattering processes. The effective model is illustrated and tested for examples of pairs of interacting point electric dipoles, where each pair can be qualitatively replaced by a model point emitter with different multipole radiation moments.

An improved ray theory and transfer matrix method‐based model for lightning electromagnetic pulses propagating in Earth‐ionosphere waveguide and its applications

AbstractAn improved ray theory and transfer matrix method‐based model for a lightning electromagnetic pulse (LEMP) propagating in Earth‐ionosphere waveguide (EIWG) is proposed and tested. The model involves the presentation of a lightning source, parameterization of the lower ionosphere, derivation of a transfer function representing all effects of EIWG on LEMP sky wave, and determination of attenuation mode of the LEMP ground wave. The lightning source is simplified as an electric point dipole standing on Earth surface with finite conductance. The transfer function for the sky wave is derived based on ray theory and transfer matrix method. The attenuation mode for the ground wave is solved from Fock's diffraction equations. The model is then applied to several lightning sferics observed in central China during day and night times within 1000 km. The results show that the model can precisely predict the time domain sky wave for all these observed lightning sferics. Both simulations and observations show that the lightning sferics in nighttime has a more complicated waveform than in daytime. Particularly, when a LEMP propagates from east to west (Φ = 270°) and in nighttime, its sky wave tends to be a double‐peak waveform (dispersed sky wave) rather than a single peak one. Such a dispersed sky wave in nighttime may be attributed to the magneto‐ionic splitting phenomenon in the lower ionosphere. The model provides us an efficient way for retrieving the electron density profile of the lower ionosphere and hence to monitor its spatial and temporal variations via lightning sferics.

Electromagnetic eigenstates and the field of an oscillating point electric dipole in a flat-slab composite structure

Electromagnetic eigenstates and the field of an oscillating point electric dipole in a flat-slab composite structure

Toroidal dipole excitations in metamolecules formed by interacting plasmonic nanorods

We show how the elusive toroidal dipole moment appears as a radiative excitation eigenmode in a metamolecule resonator that is formed by pairs of plasmonic nanorods. We analyze one such nanorod configuration - a toroidal metamolecule. We find that the radiative interactions in the toroidal metamolecule can be qualitatively represented by a theoretical model based on an electric point dipole arrangement. Both a finite-size rod model and the point dipole approximation demonstrate how the toroidal dipole moment is subradiant and difficult to excite by incident light. By means of breaking the geometric symmetry of the metamolecule, the toroidal mode can be excited by linearly polarized light and appears as a Fano resonance dip in the forward scattered light. We provide simple optimization protocols for maximizing the toroidal dipole mode excitation. This opens up possibilities for simplified control and driving of metamaterial arrays consisting of toroidal dipole unit-cell resonators.

Substrate-Induced Resonant Magnetoelectric Effects for Dielectric Nanoparticles

We study, both numerically and analytically, the effect of a substrate on the optical properties of high-refractive-index dielectric nanoparticles. We demonstrate that the optical response of subwavelength nanoparticles can be effectively modeled in the dipole approximation when the dielectric particle is replaced by a pair of point electric and magnetic dipoles with modified polarizabilities. Based on our model, we reveal the existence of the substrate-induced bianisotropic interaction (magnetoelectric coupling) between electric and magnetic dipole modes, which is strongly enhanced near the corresponding dipole resonances. This results in a specific feature of the radiation pattern that can be employed to identify the effective bianisotropic interaction experimentally.

Applicability of point-dipoles approximation to all-dielectric metamaterials

All-dielectric metamaterials consisting of high-dielectric inclusions in a low-dielectric matrix are considered as a low-loss alternative to resonant metal-based metamaterials. In this paper we investigate the applicability of the point electric and magnetic dipoles approximation to dielectric meta-atoms on the example of a dielectric ring metamaterial. Despite the large electrical size of high-dielectric meta-atoms, the dipole approximation allows for accurate prediction of the metamaterials properties for the rings with diameters up to $\ensuremath{\approx}0.8$ of the lattice constant. The results provide important guidelines for design and optimization of all-dielectric metamaterials.

Phase behavior of an amphiphilic fluid

We invoke mean-field density functional theory (DFT) to investigate the phase behavior of an amphiphilic fluid composed of a hard-sphere core plus a superimposed anisometric Lennard-Jones perturbation. The orientation dependence of the interactions consists of a contribution analogous to the interaction potential between a pair of "spins" in the classical, three-dimensional Heisenberg fluid and another one reminiscent of the interaction between (electric or magnetic) point dipoles. At fixed orientation both contributions are short-range in nature decaying as r-6 (r being the separation between the centers of mass of a pair of amphiphiles). Based upon two mean-field-like approximations for the pair correlation function that differ in the degree of sophistication we derive expressions for the phase boundaries between various isotropic and polar phases that we solve numerically by the Newton-Raphson method. For sufficiently strong coupling between the Heisenberg "spins" both mean-field approximations generate three topologically different and generic types of phase diagrams that are observed in agreement with earlier work [see, for example, Tavares et al., Phys. Rev. E 52, 1915 (1995)]. Whereas the dipolar contribution alone is incapable of stabilizing polar phases on account of its short-range nature it is nevertheless important for details of the phase diagram such as location of the gas-isotropic liquid critical point, triple, and tricritical points. By tuning the dipolar coupling constant suitably one may, in fact, switch between topologically different phase diagrams. Employing also Monte Carlo simulations in the isothermal-isobaric ensemble the general topology of the DFT phase diagrams is confirmed.

A Generalized Approach for Computation of Near Field Radiation Pattern of an Antenna

A generalized procedure in the form of an analytical formulation for the determination of radiation pattern of an antenna at any arbitrary distance which covers the near field as well as far field is presented in this paper. With the prior knowledge of either the current or field distribution on the radiating aperture, the proposed near field analysis is generic and can be applied for wide variety of antenna elements. The underlying principle of the generalized procedure is tantamount to considering the radiating aperture as an array of point electric and magnetic dipoles. The validity and novelty of the proposed new approach have been substantiated considering an open ended circular cylindrical waveguide and a conical horn as case studies and treating the far field as a special case of near field with pertinent distance criterion. The effect of change in the distance of observation ranging from reactive near field to far field on the radiation patterns of these antennas has also been discussed. The simulation studies reveal that the depicted normalized phase patterns of both the circular waveguide and conical horn follow the changes in the profile of the corresponding amplitude patterns.

Modified fundamental Airy wave

The propagation characteristics of the fundamental Airy wave are obtained; the intensity distribution is the same as that for a point electric dipole situated at the origin and oriented normal to the propagation direction. The propagation characteristics of the modified fundamental Airy wave are determined. These characteristics are the same as those for the fundamental Gaussian wave provided that an equivalent waist is identified for the Airy wave. In general, the waves are localized spatially with the peak in the propagation direction.

Lines of force of the electric point dipole

In textbooks the pictures of exact and approximate equi-potentials and the lines of force of a dimensional electric point dipole are usually presented without mentioning the equation of lines of force. Generally, these pictures are generated by numerical methods. Smythe [2] provides a special method to obtain an involved expression for it, but in this letter we show that the usual and well-known approximation to the potential due to two point charges (±q) separated by a distance a itself poses a more simply solvable problem that yields a simple expression for lines of force.