The two-body multipole problem of electrodynamics

A 2-body system composed of two objects having arbitrary distributions of charge and current is discussed. An expression for the velocity dependent potential between these two objects has been obtained in the non-relativistic approximation. This potential consists of two parts viz. a velocity independent scalar potential Φ_eff and another part which is linearly dependent on the relative velocity between the objects. The second part naturally suggests a vector potential A_eff. The potentials have been expanded into multipole terms. It has been found that Φ_eff is a sum of two components viz. Φ_EE and Φ_MM such that each multipole term in Φ_EE represents an interaction between the electric multipoles of the two systems, each term in Φ_MM represents an interaction between their magnetic multipoles whereas each term in A_eff represents an interaction between an electric multipole of one and a magnetic multipole of the other. The results have been applied to the interaction between an electric dipole and a magnetic dipole. The symmetry among the multipole terms in A_eff suggests vanishing vector potential between two identical objects. A corollary of this appears to be absence of spin orbit interaction between two identical particles in the same spin state.     

Quenching of magnetic transitions as an angular momentum effect

We investigate how the mechanisms of nuclear excitations and mesonic currents traditionally invoked to explain the quenching of low multipole magnetic transitions apply to higher spin operators. We find that there is a significant quenching of the low multipolarity spin-isospin operators by Δ-hole polarization, but that this effect decreases for higher multipoles. Conversely, the nucleon-hole polarization which has a negligible effect on low multipoles plays a predominant role in explaining the quenching of higher multipoles. Meson exchange currents provide a significant enhancement of the orbital operators but have only a slight effect on the spin components.     

Elastic charge density representation of the interaction via the nematic director field

The interaction between particle-like sources of the nematic director distortions (e.g., colloids, point defects, macromolecules in nematic emulsions) allows for a useful analogy with the electrostatic multipole interaction between charged bodies. In this paper we develop this analogy to the level corresponding to the charge density and consider the general status of the pairwise approach to the nematic emulsions with finite-size colloids. It is shown that the elastic analog of the surface electric charge density is represented by the two transverse director components on the surface imposing the director distortions. The elastic multipoles of a particle are expressed as integrals over the charge density distribution on this surface. Because of the difference between the scalar electrostatics and vector nematostatics, the number of elastic multipoles of each order is doubled compared to that in the electrostatics: there are two elastic charges, two vectors of dipole moments, two quadrupolar tensors, and so on. The two-component elastic charge is expressed via the vector of external mechanical torque applied on the particle. As a result, the elastic Coulomb-like coupling between two particles is found to be proportional to the scalar product of the two external torques and does not directly depend on the particles' form and anchoring. The real-space Green function method is used to develop the pairwise approach to nematic emulsions and determine its form and restrictions. The pairwise potentials are obtained in the familiar form, but, in contrast to the electrostatics, they describe the interaction between pairs (dyads) of the elastic multipole moments. The multipole moments are shown to be uniquely determined by the single-particle director field, unperturbed by other particles. The pairwise approximation is applicable only in the leading order in the small ratio particle size-to-interparticle distance as the next order contains irreducible three-body terms.     

Theoretical Treatment of Electric and Magnetic Multipole Radiation Near a Planar Dielectric Surface Based on Angular Spectrum Representation of Vector Field

We have developed an analytic treatment of light emission properties of electric and magnetic multipoles near a planar dielectric surface, using angular spectrum representation of vector spherical waves. The results are described in terms of spatial rotation matrix elements, so that the angular distribution of light emission for higher order multipoles is easily obtained, which enables us to evaluate basic optical near-field problems such as electric dipole radiation with arbitrary orientation with respect both to surface and observation direction. The numerical results are in good agreement with our previous experimental results and the numerical results reported by Lukosz.     

Super scattering phenomenon in active spherical nanoparticles

Localized surface electromagnetic resonances in spherical nanoparticles with gain are investigated by using the Mie theory. Due to the coupling between the gain and resonances, super scattering phenomenon is raised and the total scattering efficiency is increased by over six orders of magnitude. The dual frequency resonance induced by the electric dipole term of the particle is observed. The distributions of electromagnetic field and the Poynting vector around nanoparticles are provided for better understanding different multipole resonances. Finally, the scattering properties of active spherical nanoparticles are investigated when the sizes of nanoparticles are beyond the quasi-static limit. It is noticed that more highorder multipole resonances can be excited with the increase of the radius. Besides, all resonances dominated by multipole magnetic terms can only appear in dielectric materials.     

On the Frenkel problem of equivalent steady currents in a sphere

The Frenkel problem of substituting the 3D system of steady currents given in one of two concentric spherical regions by an equivalent system of currents (i.e., by that inducing the same external magnetic field) that is distributed over the surface of the other region is considered. A method of multipole moments providing the direct solution (without calculating the fields) of the problem is described. The case of currents with the density components represented by cubic polynomials of the Cartesian coordinates is considered as an example.     

Multipole vector solitons in nonlocal nonlinear media

We show that multipole solitons can be made stable via vectorial coupling in bulk nonlocal nonlinear media. Such vector solitons are composed of mutually incoherent nodeless and multipole components jointly inducing a nonlinear refractive index profile. We found that stabilization of the otherwise highly unstable multipoles occurs below certain maximum energy flow. Such a threshold is determined by the nonlocality degree.     

Multipole analysis of a generic system of dielectric cylinders and application to fibrous materials

A multipole theory describing the interactions between dielectric cylinders in a uniform field is developed. We treat the most general case of $N$ parallel cylinders placed in arbitrary positions. The exact theory is obtained by developing the polarisation charge surface density on each cylinder in a Fourier series. The related coefficients, the so-called multipoles, may be obtained from a linear set of equations which is derived and analysed in the paper. For systems of closely spaced cylinders, with high ratio of the dielectric constant of the cylinders compared to that of the homogeneous medium (in the worst case, conductive cylinders in contact with each other) a very large number of multipole terms is required to achieve convergence. In spite of the large number of required terms, the general multipole expansion is rapidly convergent in all other cases and is important from a theoretical point of view. Numerical results are presented for canonical dispositions of cylinders and for more complicated arrangements. Finally, such a multipole expansion has been applied to the dielectric characterisation of composite materials formed by a regular array of parallel cylinders, thereby obtaining the equivalent permittivity using a numerically efficient technique.     

An algorithm for the calculation of the net induction energy and induced multipole tensors in a set of charge distributions with non-linear contributions of all potentials of external fields and both permanent and induced multipoles

This algorithm for the calculation of the induced multipole tensors in a set of charge distributions includes the contributions of partial derivatives of arbitrary order of the potentials defined by both induced and permanent multipoles as well as non-uniform external fields of arbitrary strength. Specific equations are given for both systems with and without translational order. The algorithm, which is based on the Maxwell invariant form, uses direct extensions of algorithms previously developed and tested for the calculation of permanent multipole energies and induced dipole vectors (when the non-uniformity of the field is neglected). The induced tensor components are calculated iteratively. The first approximation, which gives the components as the solution of a set of simultaneous linear equations, includes all nonlinear, non-uniform contributions of permanent multipoles and external fields as well as all contributions linear in derivatives of the induced potentials. The induced tensor components are then used to calculate the net induction energy. The general relations between the polarizability tensors with respect to a centre and the moments of the polarizability densities about the centre are derived.     

On a hollow superconducting sphere in a magnetic field

A system consisting of a single-domain ferromagnetic ball surrounded by a spherical superconducting shell and current-carrying wires wound on the shell is considered. The magnetic moment of the superconductor is calculated in terms of the London theory, in particular, for the case when the sum of the magnetic moments of the ball and system of currents is equal to a null vector.     

On the simplest equivalent currents of an ellipsoidal body

The solution is obtained to the Frenkel problem of determining the surface current on the boundary of an ellipsoidal region of space, equivalent (i.e., producing the same external magnetic field) to the simplest steady-state current specified in the bulk of this region. Simultaneously, multipole representations of pseudoscalar magnetic potentials of the volume and surface currents are determined in the framework of this particular problem. It is shown that these representations are completely analogous to the corresponding multipole representations of the scalar potentials of the ellipsoid associated with the volume or surface distributions of scalar sources (charges or masses).     

Nuclear polarization in muonic atoms of deformed nuclei

The treatment of nuclear polarization correction in muonic atoms belonging to deformed nuclei is analyzed. The geometrical factors involved are expanded into a series of multipoles and the exact expansion coefficients are calculated. It is shown that, using reasonable assumptions about the nuclear spectrum, the nuclear polarization correction may be expressed as a shift of all hyperfine components plus a renormalization of the even multipole hyperfine interaction constants. All nuclear excited states contribute to the shift, but the ground-state rotational band gives an over-whelming contribution to the multipole moment renormalization. The effect of the ground-state band is analyzed in detail. The radial coefficients are calculated and an approximate formula, applicable over a broad range of atomic numbers and deformations, is obtained. By comparing our results with exact calculations we conclude that this part of the nuclear polarization correction may be calculated with accuracy better than 10 %.     

Magnetic multipole moments

Literature definitions of magnetic multipole moment operators are shown to be at variance, and new definitions are formulated which are consistent with a general multipole interaction hamiltonian and with the radiation field of a dynamic charge distribution. The applicability of traceless multipole moments is examined.

The multipole hamiltonian is used to derive expressions for some magnetic quadrupole distortion tensors. For those describing the quadrupole moment induced by a magnetic field and by a field gradient the number of independent components for various molecular symmetries is evaluated.     

Collective Multipole Expansions and the Perturbation Theory in the Quantum Three-Body Problem

The perturbation theory with respect to the potential energy of three particles is considered. The first-order correction to the continuum wave function of three free particles is derived. It is shown that the use of the collective multipole expansion of the free three-body Green function over the set of Wigner D-functions can reduce the dimensionality of perturbative matrix elements from twelve to six. The explicit expressions for the coefficients of the collective multipole expansion of the free Green function are derived. It is found that the S-wave multipole coefficient depends only upon three variables instead of six as higher multipoles do. The possible applications of the developed theory to the three-body molecular break-up processes are discussed.     

Multipole analysis of the radiation from near-field optical probes

We experimentally and theoretically analyze the radiation emitted from subwavelength-sized apertures in near-field optical probes. By decomposing the experimentally obtained radiation patterns into vector spherical waves, we describe the fields in terms of a series of multipole sources. We fit polarization-resolved angular intensity distributions, measured as far as 150 degrees from the normal, with dipole, quadrupole, and octupole radiation. We find that the magnetic and the electric dipole components are dominant but that the interference terms between dipoles and higher-order poles are not negligible. This result can be used as the basis for understanding near-field optical interactions and images.     

Surface integral method to determine guided modes in uniaxially anisotropic embedded waveguides

A surface integral method is presented to calculate the eigenmodes of an uniaxially anisotropic embedded channel waveguide. The electromagnetic field components are expressed with electric and magnetic vector potentials which are parallel with the optic axis and for which the scalar Helmholtz-equations hold using surface integral representation and the Green's functions of the vector potentials. The single component vector potentials are expanded in Fourier-series on the internal side of the dielectric interface. The Fourier-coefficients and the corresponding eigenvalues are obtained by minimizing the quadratic difference of the longitudinal field components in the cladding and core along the dielectric interface. The convergence of the series expansion is examined numerically and the eigenvalues obtained with the surface integral method agree with those obtained by Finite Element Method up to 5 significant digits.     

Non-radiative decay of a dipole emitter close to a metallic nanoparticle: Importance of higher-order multipole contributions

The contribution of higher-order multipoles to radiative and non-radiative decay of a single dipole emitter close to a spherical metallic nanoparticle is re-examined. Taking a Ag spherical nanoparticle (AgNP) with the radius of 5 nm as an example, a significant contribution (between 50% and 101% of the total value) of higher-order multipoles to non-radiative rates is found even at the emitter distance of 5 nm from the AgNP surface. On the other hand, the higher-order multipole contribution to radiative rates is negligible. Consequently, a dipole-dipole approximation can yield only an upper bound on the apparent quantum yield. In contrast, the non-radiative rates calculated with the quasistatic Gersten and Nitzan method are found to be in much better agreement with exact electrodynamic results. Finally, the size corrected metal dielectric function is shown to decrease the non-radiative rates near the dipolar surface plasmon resonance.     

Contribution from cosmological scalar perturbations to the angular velocity spectrum of extragalactic sources

The possibility of the influence of adiabatic scalar perturbations on the angular velocity spectrum of extragalactic sources is considered. The multipole expansion coefficients of the angular velocity field in terms of vector spherical harmonics are calculated. We show that there is no contribution from adiabatic perturbations to the angular spectrum for a spatially flat Universe at the dusty stage, while there is a contribution only to the electric multiple coefficients at the stage of ??-term domination. The cases of long-wavelength and short-wavelength perturbations are considered separately. The relationship between the multipole angular velocity spectrum and the primordial scalar perturbation spectrum is discussed.