Multipole Series Research Articles

Higher-order multipoles in metaphotonic structures: reviewing symmetric approaches and constructing a comprehensive theoretical method

This research explores localized electric fields in two-dimensional (2D) photonic crystals, focusing on higher-order multipoles. Novel formulas are developed to accurately describe and analyze these complex electromagnetic modes. By considering various crystal element arrangements, we investigate the formation and interactions of electric, magnetic, and toroidal moments. We examine the effects of symmetry on the localized electromagnetic fields in 2D periodic structures and explain how bound states in the continuum (BICs) arise and behave in different symmetry classes. It is important to point out that every singularity point that occurs in multipolar series expansions depends on a BIC point, and it inspires us to study the characteristics of them. Furthermore, we analyze a variety of BICs and use perturbed and degeneracy breaking to investigate high-Q-factor quasi-BICs. The results provide a thorough understanding of how symmetry and multipolar interactions affect potential applications in cutting-edge metasurfaces and photonic devices (metaphotonic).

Convergence of the Laplace and the alternative multipole expansion approximation series for the Coulomb potential

Multipole expansion is a powerful technique used in many-body physics to solve dynamical problems involving correlated interactions between constituent particles. The Laplace multipole expansion series of the Coulomb potential is well established in literature. We compare its convergence with our recently developed perturbative and analytical alternative multipole expansion series of the Coulomb potential. In our working, we confirm that the Laplace and the analytical alternative multipole expansion series are equivalent as expected. In terms of performance, the perturbative alternative multipole expansion series underapproximate the expected results to some extent while the Laplace and the analytical alternative multipole expansion series yield results which are relatively accurate but oscillatory in nature even with a higher number of angular momentum terms employed. As a practical example, we have evaluated the Slater double integrals for two-electron systems using the multipole expansion techniques and a mean field approximation. The estimated results show that only spherical interactions are dominant while the higher-order interactions are negligible. To highlight the discrepancy in the application of each of the formulations of the multipole expansion series for the electron-electron interaction potential, an estimation of the non-relativistic groundstate energies of some helium-like systems, evaluated using the spherical approximation of the multipole potential, is provided. Our findings are likely to be useful in the treatment of the Coulomb potential in electronic structure calculations as well as in celestial mechanics.

Fundamental Solutions of the Equations of the Oscillation Theory for Anisotropic Elastic Media

The construction of fundamental solutions in R3 for the equations of harmonic vibrations in the theory of elasticity of anisotropic elastic media is carried out. Solutions are constructed in the form of multipole series. Theorems on the convergence of series in the topology of compact convergence in R3/0 are proved. The problems on constructing some singular solutions of the theory of vibrations of an anisotropic body are discussed. The fundamental solution of the oscillation equations for an isotropic medium is obtained in a closed form.

Modified multipoles in photonics

Multipole decomposition method is a promising tool for investigation of radiating or scattering responses of electromagnetic sources or particles. It works even in the case of relatively complicated and compound scatterers like multilayer particles, clusters, or asymmetrical systems. Commonly, the radiation fields of point electric or magnetic sources are decomposed only into electric or magnetic dipole moments, while real sources can be described by a series of multipoles, including higher multipoles and toroidal moments. In this paper, we introduce the concept of modified multipoles describing real sources of electric, magnetic, and toroidal types. Using the analytical expressions of first-order multipoles, we discuss how they depend on the position of the center of radiation, as well as on the shift of the source, relative to the center of coordinates. We present results of multipoles for the sources with defects and asymmetry. The long-awaited question about distinguishing radiation patterns of electric and toroidal dipole moments in a far-field zone is discussed and solved in this paper. We show that radiation patterns of shifted electric and toroidal dipoles can be rotated to different angles relative to each other due to shifting. Moreover, we discuss modified anapoles. Our modified dipole approach will be useful for multipole analysis of complex systems in photonics such as nanoparticle clusters, metamaterials, and nanoantennas, as well as for better understanding issues of toroidal electrodynamics.

Reflection of the individual multipoles fields from a flat substrate in the angular spectrum approach

In this report, we give the derivation of the electromagnetic field of multipoles over a substrate, using the angular spectrum representation. The multipole series up to the electric quadrupole is considered. If necessary, this approach can easily be extended to multipoles of any order. As a result, analytical expressions for the scattered and reflected multipole fields as an infinite sum of plane waves of different polarizations are obtained.

Symmetry-dependent analytical all-electron potential for helium atom

Electron–electron interaction is the origin of the many-body problems usually encountered in physics and chemistry. Helium atom and other two-electron systems are the simplest many-body systems in nature. The Schrödinger equation even for such simple systems cannot be solved exactly without resorting to approximate methods. In this study, we have suggested a symmetry-dependent analytical all-electron potential for helium atom derived using an alternative multipole expansion, a variational technique, and a mean-field approximation. We have calculated the non-relativistic groundstate energy for helium atom to be −2.90422284. The suggested all-electron potential has a local Coulomb potential with embedded nuclear charge screening effect in the leading term of the multipole potential. A non-local component of the potential emanates from the higher-order angular momentum terms of the multipole series expansion. The higher-order multipole interactions are fully included through the exchange correlation processes where the interacting electrons exchange their angular momenta via the operator. With the derived potential, the effects of the local long-range and non-local short-range components, and the finite nuclear mass corrections are tested. Our results are in reasonable agreement with literature values. Indeed with the finite nuclear mass correction, we obtain the groundstate energy of helium atom to be −2.90382769.

Multipole infinite series expansion of the Coulomb repulsion term using analytical spherical Bessel-like functions

Special functions are particular mathematical functions which arise in the solutions of various classical problems in physics. The functions are quite useful in mathematical analysis. Mostly, they find applications in applied mathematics, theoretical physics, and engineering. Special functions are usually uniquely described by a generating function, a Rodrigues’ formula, and a recurrence relation. In this work, we derive analytical spherical Bessel-like functions corresponding to the infinite degree power series expansion of the functions obtained in the multipole series expansion of the Coulomb repulsion term. The convergence of the truncated power series expansion of the functions with the corresponding derived analytical functions, as well as the comparison of analytical functions of the first and second kinds, is analyzed. Rodrigues’ formula and a recurrence relation for obtaining these analytical functions are also presented. It is shown that the higher order even and odd spherical Bessel-like functions of the first kind are derivable from the lowest order functions of the first and second kinds, respectively.

Atomistic vs. continuum models of nanoporous elastic solid: Stress fields, size-dependent effective stiffness and surface constants

The method of classical molecular dynamics (MD) is applied to study the stress fields and effective elastic stiffness tensor of nanoporous materials. A continuum model for a nanoporous solid involving a surface layer of sub-nanometer thickness is proposed. Systematic comparisons of the unit cell atomistic MD results are performed with two continuum models, one based on the finite element analysis and another based on the multipole series expansion. This comparison validates the continuum models, shows their applicability limits, and provides reliable identification of the surface elastic constants.

Piecewise Multipole-Expansion Implicit Solvation for Arbitrarily Shaped Molecular Solutes.

The multipole-expansion (MPE) model is an implicit solvation model used to efficiently incorporate solvent effects in quantum chemistry. Even within the recent direct approach, the multipole basis used in MPE to express the dielectric response still solves the electrostatic problem inefficiently or not at all for solutes larger than approximately ten non-hydrogen atoms. In existing MPE parametrizations, the resulting systematic underestimation of the electrostatic solute-solvent interaction is presently compensated for by a systematic overestimation of nonelectrostatic attractive interactions. Even though the MPE model can thus reproduce experimental free energies of solvation of small molecules remarkably well, the inherent error cancellation makes it hard to assign physical meaning to the individual free-energy terms in the model, raising concerns about transferability. Here we resolve this issue by solving the electrostatic problem piecewise in 3D regions centered around all non-hydrogen nuclei of the solute, ensuring reliable convergence of the multipole series. The resulting method thus allows for a much improved reproduction of the dielectric response of a medium to a solute. Employing a reduced nonelectrostatic model with a single free parameter, in addition to the density isovalue defining the solvation cavity, our method yields free energies of solvation of neutral, anionic, and cationic solutes in water in good agreement with experiment.

Interaction of circularly polarized light with an absorptive electromagnetic conductor sphere – Radiation force and spin torque

The concept of a perfect electromagnetic conductor (PEMC) material, which is a class of a magneto-electric metamaterial, has been recently examined in the context of the optical radiation force theory [J. Quant. Spectr. Rad. Transfer, 233 (2019) 21–28; ibid 256 (2020) 107280], with particular emphasis on the rotary polarization effect and the contribution of the cross-polarized waves to the transfer of linear momentum, assuming an illumination of linearly-polarized plane progressive waves. The present investigation shows that additional cross-interference factors, which cannot be ignored, contribute to the time-averaged optical radiation force exerted on an absorptive electromagnetic conductor (AEMC) sphere when the incident field becomes circularly-polarized. Moreover, the analysis is extended to investigate the co-polarized, cross-polarized and cross-interference components of the time-averaged optical spin radiation torque. The multipole series expansion method in spherical coordinates is utilized to derive exact mathematical partial-wave series for the co-polarized, cross-polarized and cross-interference components of the radiation force and torque based on the integration of Maxwell’s radiation stress tensor and its moment over a spherical surface of large radius enclosing the sphere suspended in a lossless medium of wave propagation. Numerical results for the radiation force and spin torque efficiencies and their corresponding co-polarized, cross-polarized and cross-interference components illustrate the analysis with emphasis on the size parameter of the sphere ka, its electromagnetic admittance M as well as its imaginary part amounting to absorption. The computations are of importance from the standpoint of the fundamentals of the linear and angular momenta of circularly-polarized electromagnetic waves transferred to an AEMC and related applications in optical tweezers and particle manipulation of objects exhibiting circular dichroism.

Ellipsoidal inhomogeneity with anisotropic incoherent interface. Multipole series solution and application to micromechanics

The ellipsoidal inhomogeneity with transversely isotropic incoherent interface is considered in the conductivity and elasticity context. The general imperfect interface is modeled by the first order approximation of thin transversely isotropic interphase layer. This model expands the Gurtin et al. (1998) theory of curved deformable interfaces in solids with a nanometer-scale microstructure to the incoherent interfaces between the dissimilar elastic materials and provides a certain insight into the interface moduli. The rigorous analytical solution to the conductivity and elasticity problems has been obtained by the multipole expansion method in terms of ellipsoidal solid harmonics. An accurate fulfillment of the imperfect interface conditions reduces the model boundary value problem to the linear algebraic system for multipole strengths. These results apply equally to the inhomogeneities with anisotropic interphases and nano level incoherent interfaces. The obtained solutions are valid for the non-uniform far loading and are readily incorporated in the many-particle (finite cluster or representative unit cell) model of heterogeneous solid with anisotropic incoherent interface. The tensors of effective conductivity and elastic stiffness of composite with ellipsoidal inhomogeneities are evaluated using Maxwell homogenization scheme. The obtained accurate numerical data indicate that the effective properties of composite may vary widely due to shape of inhomogeneities and the interface anisotropy ratio. Taking the incoherency of interface into account may increase reliability of predicting the behavior of nanostructured solids.

Method of image charges for describing deformation of bounded two-dimensional solids with circular inclusions.

We present a method for predicting the linear response deformation of finite and semi-infinite 2D solid structures with circular holes and inclusions by employing the analogies with image charges and induction in electrostatics. Charges in electrostatics induce image charges near conductive boundaries and an external electric field induces polarization (dipoles, quadrupoles, and other multipoles) of conductive and dielectric objects. Similarly, charges in elasticity induce image charges near boundaries and external stress induces polarization (quadrupoles and other multipoles) inside holes and inclusions. Stresses generated by these induced elastic multipoles as well as stresses generated by their images near boundaries then lead to interactions between holes and inclusions and with their images, which induce additional polarization and thus additional deformation of holes and inclusions. We present a method that expands induced polarization in a series of elastic multipoles, which systematically takes into account the interactions of inclusions and holes with the external field, between them, and with their images. The results of our method for linear deformation of circular holes and inclusions near straight and curved boundaries show good agreement with both linear finite element simulations and experiments.

Multipole decomposition of redshift drift: Model-independent mapping of the expansion history of the Universe

We consider redshift drift in a general space-time as expressed in terms of physically interpretable multipole series. An important realisation from the derived results is that redshift drift cannot in general be thought of as a direct probe of the average expansion rate of the Universe due to the presence of structure along the light beams from the astrophysical sources to the observer. We also predict the general presence of a dipolar and a quadrupolar offset in the detection of redshift drift for observers placed in locally anisotropic environments.

A fully general, non-perturbative treatment of impulsive heating

ABSTRACT Impulsive encounters between astrophysical objects are usually treated using the distant tide approximation (DTA) for which the impact parameter, b, is assumed to be significantly larger than the characteristic radii of the subject, rS, and the perturber, rP. The perturber potential is then expanded as a multipole series and truncated at the quadrupole term. When the perturber is more extended than the subject, this standard approach can be extended to the case where rS ≪ b < rP. However, for encounters with b of order rS or smaller, the DTA typically overpredicts the impulse, Δv, and hence the internal energy change of the subject, ΔEint. This is unfortunate, as these close encounters are the most interesting, potentially leading to tidal capture, mass stripping, or tidal disruption. Another drawback of the DTA is that ΔEint is proportional to the moment of inertia, which diverges unless the subject is truncated or has a density profile that falls off faster than r−5. To overcome these shortcomings, this paper presents a fully general, non-perturbative treatment of impulsive encounters which is valid for any impact parameter, and not hampered by divergence issues, thereby negating the necessity to truncate the subject. We present analytical expressions for Δv for a variety of perturber profiles, apply our formalism to both straight-path encounters and eccentric orbits, and discuss the mass-loss due to tidal shocks in gravitational encounters between equal-mass galaxies.

Relations between the Phases and Amplitudes of Multipole Components of an Acoustic Field Scattered by Discrete Inhomogeneities

An acoustic field scattered by an object can be presented as a multipole series. The set of values of every coefficient in this series is limited for objects with cylindrical symmetry when there are no internal sources of energy. Relations are presented for the phases and amplitudes of these coefficients, and between the powers of the absorbed and scattered fields.

Optical radiation force on a perfect electromagnetic conductor (PEMC) sphere

This work extends the scope of the previous study on the electromagnetic radiation force on a perfect electromagnetic conductor (PEMC) cylinder [F.G. Mitri, J. Quant. Spectr. Rad. Transfer, 233 (2019) 21–28] to the case of a sphere of arbitrary size, illuminated by linearly-polarized plane progressive waves. In contrast with the PEMC cylinder case, the analysis shows that the effect of the cross-polarized waves compensates that of the co-polarized waves such that the optical radiation force is unaffected by the change of the electromagnetic admittance parameter M of the sphere. The multipole series expansion method in spherical coordinates is used to derive exact mathematical series for the co-polarized and cross-polarized components of the longitudinal radiation force. Numerical results for the radiation force efficiency and its co-polarized and cross-polarized components demonstrate the individual contributions. The results show that the cross-polarized component of the radiation force efficiency can be positive or negative as the dimensionless size ka varies. Moreover, it vanishes for ka ≈ 1.78 regardless of the admittance parameter of the PEMC sphere. Notice that the total force (i.e. the sum of the co-polarized and cross-polarized components) is always repulsive (i.e., positive). It is also verified that the results are in complete agreement with the law of energy conservation applied to scattering, where the extinction and scattering energy efficiencies are equal, and independent of M. Furthermore, an equivalent expression for the longitudinal radiation force efficiency is derived stemming from an analysis of the asymmetry parameter and the law of energy conservation applied to scattering. Additional computations for the co-polarized and cross-polarized components of the scattering asymmetry parameter are presented and discussed. The results are of some importance from the standpoint of the basics of the electromagnetic radiation force theory and related applications in particle manipulation.

Description of heterogeneous plasma microfield and optical properties of plasma by the QUIP model

Optical properties of plasmas are determined by presence of fluctuating microscopic electric field. The present work provides thorough analysis of contemporary models and points out their shortcomings. To overcome the latter, we take the QUIP (QUasi-Independent Particles) model derived ab initio. We provide generalization of the model allowing to account for microfield heterogeneity up to octupole term. We investigate convergence of the multipole series and show that higher order terms can be neglected. The model does not require laborious computations because all formulae are rather simple and are given in explicit form. This fact is an advantage of the proposed model compared to other contemporary models. We perform verification of the model via comparison with experiments. We emphasize that the comparison should be made with respect to the number of observed lines because this number strongly depends on the selected model. We outline experiments suitable for such testing. These are the experiments on Ar+Kr radiating plasma heated by laser radiation. In these experiments, the entire Ar+16 spectral series is observed. The QUIP model correctly describes the number of observed lines of the series, so its adequateness is justified.

Preconditioner for estimation of multipole sources via full waveform inversion

Accurate representation and estimation of seismic sources is crucial to the joint medium-source full waveform inversion problem. We focus on the source estimation subproblem and its difficulties, where seismic sources are modeled by truncated series of multipoles. The source full waveform inversion formulation results in a highly ill-conditioned (potentially ill-posed) linear least squares problem which we attempt to solve iteratively via conjugate gradient. Our main contribution lies in developing a preconditioner to accelerate the performance of conjugate gradient on multipole source inversion. The proposed preconditioner consists of (fractional) time derivative/integral operators based on analytical solutions to the wave equation with multipole sources in an unbounded, homogeneous medium. Numerical results in 2D demonstrate that the conjugate gradient iterations are accelerated when incorporating the proposed preconditioning scheme.

Black-hole scattering with general spin directions from minimal-coupling amplitudes

We study the link between classical scattering of spinning black holes and quantum amplitudes for massive spin-$s$ particles. Generic spin orientations of the black holes are considered, allowing their spins to be deflected on par with their momenta. We rederive the spin-exponentiated structure of the relevant tree-level amplitude from minimal coupling to Einstein's gravity, which in the $s\to\infty$ limit generates the black holes' complete series of spin-induced multipoles. The resulting scattering function is seen to encode in a simple way the known net changes in the black-hole momenta and spins at first post-Minkowskian order. We connect our findings to a rigorous framework developed elsewhere for computing such observables from amplitudes.

Dispersion Forces: Neither Fluctuating Nor Dispersing

Dispersion forces are ubiquitous in chemistry, yet they are often misunderstood. This article provides background into why they are called dispersion forces and explains how to describe them in terms of time-independent quantum mechanics. The article also describes the breakdown of the multipole series that is often used to describe dispersion forces, and it comments on the representation of dispersion forces in molecular mechanics. In general, the article tries to provide a proper framework for understanding and describing dispersion forces.