Number Of Multipoles Research Articles

Creating exact multipolar fields with azimuthally modulated rf cavities

Radio-frequency cavities used in modern particle accelerators operate in ${\mathrm{TM}}_{m10}$-like modes composed of a single, dominant multipole of order $m$; $m=0$ modes are used for the longitudinal acceleration of a particle beam and $m\ensuremath{\ne}0$ modes for controlling transverse beam dynamics. The practical design of the latter, however, can be complex and require extensive analysis through the iteration of both approximate mathematical models and computationally expensive simulations to optimize the performance of the structure. In this paper we present a new, systematic method for designing azimuthally modulated rf cavities that support modes composed of any number and magnitude of user-specified transverse multipoles, either with or without a longitudinally accelerating component. Two case studies are presented of rf cavity designs that support modes composed of a longitudinally accelerating field in addition to a single transverse multipole, and designs that support modes composed of two transverse multipoles. We discuss generalizing the discoveries and conclusions from the two case studies to designing cavities that support modes composed of any number of multipoles. The theoretical work is verified with analysis of 3D simulations and experimental measurements are presented of a cavity operating in a 3 GHz mode that simultaneously longitudinally accelerates and transversely focuses a beam.

Applicability of multipole decomposition to plasmonic- and dielectric-lattice resonances.

Periodic nanoparticle arrays have attracted considerable interest recently since the lattice effect can lead to spectrally narrow resonances and tune the resonance position in a broad range. Multipole decomposition is widely used to analyze the role of the multipoles in the resonance excitations, radiation, and scattering of electromagnetic waves. However, previous studies have not addressed the validity and accuracy of the multipole decomposition around the lattice resonance. The applicability of the exact multipole decomposition based on spherical harmonics expansion has not been demonstrated around the lattice resonance with the strong multipole coupling. This work studies the two-dimensional periodic arrays of both plasmonic and dielectric nanospheres and compares the multipole decomposition results with the analytic ones around their lattice resonances. We study both the effective polarizabilities of multipoles and the scattering spectra of the structures. The analytical results are calculated from the coupled dipole-quadrupole model. This study demonstrates that the exact multipole decomposition agrees well with the numerical simulation around lattice resonances. Only a small number of multipoles are required to represent the results accurately.

The Effect of Multipoles on the Elasto-Plastic Properties of a Crystal: Theory and Three-Dimensional Dislocation Dynamics Modeling

Abstract Plastic deformation in metals is dominated by the interactions among dislocations and other defects inside the crystal. A large number of dislocation multipoles (dipoles, tripoles, quadrupoles, etc.) can form during plastic deformation. Depending on the relative position and the orientation of the dislocations, interactions in and between multipoles can change the elastoplastic properties of a material. The authors of this article investigate the effect of dislocation multipoles on the elastoplastic properties of a material. This is performed analytically under different multipole configurations (i.e., the distance between active glide planes and the signs of the dislocations) as well as using a three-dimensional discrete dislocation dynamics (DDD) code. The simulations show that multipoles exhibit a hardening/softening effect when the sign of the dislocations involved is the same, and a hardening effect only when the dislocations are of opposite sign to nearby ones. The distance between the two neighboring dislocations was also affecting the proportional limit (PL) for the material. Such hardening or flow stress (FS) results, as in this study, can be incorporated into larger-scale modeling work.

Diffraction of electromagnetic waves by an extended gravitational lens

We continue our study of the optical properties of the solar gravitational lens (SGL). Taking the next step beyond representing it as an idealized monopole, we now characterize the gravitational field of the Sun using an infinite series of multipole moments. We consider the propagation of electromagnetic (EM) waves in this gravitational field within the first post-Newtonian approximation of the general theory of relativity. The problem is formulated within the Mie diffraction theory. We solve Maxwell's equations for the EM wave propagating in the background of a static gravitational field of an extended gravitating body, while accounting for multipole contributions. Using a wave-theoretical approach and the eikonal approximation, we find an exact closed form solution for the Debye potentials and determine the EM field at an image plane in the strong interference region of the lens. The resulting EM field is characterized by a new diffraction integral. We study this solution and show how the presence of multipoles affects the optical properties of the lens, resulting in distinct diffraction patterns. We identify the gravitational deflection angle with the individual contributions due to each of the multipoles. Treating the Sun as an extended, axisymmetric, rotating body, we show that each zonal harmonics causes light to diffract into an area whose boundary is a caustic of a particular shape. The appearance of the caustics modifies the point-spread function (PSF) of the lens, thus affecting its optical properties. The new wave-theoretical solution allows the study gravitational lensing by a realistic lens that possesses an arbitrary number of gravitational multipoles. This {\em angular eikonal method} represents an improved treatment of realistic gravitational lensing. It may be used for a wave-optical description of many astrophysical lenses.

Generalized hybrid anapole modes in all-dielectric ellipsoid particles [Invited

Numerous exciting optical effects in all-dielectric high-refractive-index structures are associated with so-called toroidal electrodynamics. Among these effects are anapoles, nonradiated states caused by interference phenomena, e.g. between electric dipole and toroidal dipole modes. For a spherical particle it is possible to reach simultaneous destructive interference for electric, magnetic, and corresponding toroidal dipole modes (so-called hybrid anapole mode), by varying the refractive index and/or particle size. However, there are no sufficient degrees of freedom within spherical geometry to extend the hybrid anapole mode effect to higher multipoles. Due to the optical theorem, it is also impossible to create the ideal anapole with destructive interference for all multipoles under plane wave illumination. In principle, it is possible to suppress radiation losses for the finite number of multipoles only by constructing the nanoantenna with complex geometry. Our approach of the hybrid anapole state excitation, we demonstrate in ellipsoidal all-dielectric particle providing cancellation of both electric and magnetic scattering up to quadrupole modes. This effect is achieved due to the optimised geometry of the ellipsoidal particle. Moreover, we provide classification of novel anapoles arising due to interference between moments and their mean- square radii (MSR) of electric, magnetic and toroidal family and introduce generalized anapoles for high order interaction between moments. Our concept is useful for the design of light controlling devices, reflectionless metasurfaces, high Q-factor opened resonators and nonscattering particle development.

Multipole decompositions for directional light scattering

Applications of the multipole decomposition method for investigations of directional light scattering by a single nanoparticle and nanoparticle structures located in a finite spatial region are discussed. It is shown that, even in the case of relatively large scatterers, the multipole decomposition obtained in the long-wavelength approximation (LWA) may provide much better convergence than the multipole decomposition with the exact multipoles obtained from the spherical harmonics expansion. For an explanation of this seeming paradox, we derive in real space the exact multipole decomposition based on the spherical harmonics, presenting exact expressions for multipoles up to the electric 16-pole. Results obtained with the exact and approximate multipole expressions are discussed and compared. It is shown that for shape-anisotropic finite-size scatterers with different geometrical dimensions (like plates, rods, disks, rings, etc.), the required number of approximate multipoles providing accurate results may be much smaller than the required number of exact multipoles. For applicability of the LWA multipole decomposition, the only important parameter is the small ratio of the scatter size (its projection) in the scattering direction to the light wavelength. If this condition is fulfilled, the multipole decomposition with a small number of LWA multipoles is simpler than that based on the exact multipoles.

Covariant equations of motion of extended bodies with arbitrary mass and spin multipoles

This paper employs the post-Newtonian approximations of scalar-tensor theory of gravity along with the Cartesian STF tensors and the Blanchet-Damour multipole formalism to derive translational and rotational equations of motion of N extended bodies with arbitrary distribution of mass and velocity. We assume that spacetime can be covered by a global coordinate chart which is Minkowskian at infinity. We also introduce N local coordinate charts adapted to each body and covering a finite domain of space around the body. Gravitational field of each body is parametrized by an infinite set of the body's mass and spin multipoles and tidal multipoles of external N-1 bodies. The origin of the local coordinates is set moving along accelerated worldline of the center of mass of the body by an appropriate choice of the internal and external dipole moments of its gravitational field. Translational and rotational equations of motion are derived by integrating microscopic equations of matter and applying the method of asymptotic matching. The matching is also used for separating the post-Newtonian self-field effects from the external gravitational environment and for constructing the effective background spacetime manifold. It allows us to present the equations of translational and rotational motion of each body in covariant form by making use of the Einstein principle of equivalence. Our approach significantly generalizes the Mathisson-Papapetrou-Dixon covariant equations of motion with regard to the number of body's multipoles and the post-Newtonian terms taken into account. The equations of translational and rotational motion derived in the present paper include the infinite set of mass and spin multipoles of the bodies and can be used for much more accurate prediction of orbital dynamics of extended bodies in inspiraling binary systems before they merge.

COMPARING CORONAL AND HELIOSPHERIC MAGNETIC FIELDS OVER SEVERAL SOLAR CYCLES

ABSTRACT Here we use the PFSS model and photospheric data from Wilcox Solar Observatory, SOHO/MDI, SDO/HMI, and SOLIS to compare the coronal field with heliospheric magnetic field measured at 1 au, compiled in the NASA/NSSDC OMNI 2 data set. We calculate their mutual polarity match and the power of the radial decay, p, of the radial field using different source surface distances and different number of harmonic multipoles. We find the average polarity match of 82% for the declining phase, 78%–79% for maxima, 76%–78% for the ascending phase, and 74%–76% for minima. On an average, the source surface of 3.25 R S gives the best polarity match. We also find strong evidence for solar cycle variation of the optimal source surface distance, with highest values (3.3 R S ) during solar minima and lowest values (2.6 R S –2.7 R S ) during the other three solar cycle phases. Raising the number of harmonic terms beyond 2 rarely improves the polarity match, showing that the structure of the HMF at 1 au is most of the time rather simple. All four data sets yield fairly similar polarity matches. Thus, polarity comparison is not affected by photospheric field scaling, unlike comparisons of the field intensity.

A Point-Adaptive Grouping Scheme of MLFMA for Electrically Large Scattering Simulation

A Gaussian point-adaptive grouping scheme for the multilevel fast multipole algorithm (MLFMA) has been proposed in this communication to significantly reduce the heavy memory cost resulted from using the basis functions defined on large patches without sacrificing the accuracy of the numerical solutions. The grouping process in the MLFMA has been considered as a single-objective optimization problem and is solved by using the clustering algorithm for Gaussian quadrature points on each patch. Meanwhile, the constraint of the addition theorem for MLFMA is still satisfied. As a result, the presented scheme is able to acquire an optimal number of multipoles for each basis function. Compared to the conventional octree grouping scheme or other grouping schemes used in the past, the method in this communication is advantageous in terms of the memory efficiency and the solution accuracy. Numerical examples have been given to demonstrate the validity and effectiveness of the proposed scheme.

Hidden-order symmetry and superconductivity inURu2Si2investigated by quasiparticle interference

The hidden order (HO) in URu2Si2 has been determined as a high rank multipole formed by itinerant 5f-electrons with distinct orbital structure imposed by the crystalline electric field. Because this can lead to a considerable number of different multipoles it is of great importance to use microscopic techniques that are sensitive to their subtle physical differences. Here we investigate whether quasiparticle interference (QPI) method can distinguish between the two most frequently proposed HO parameter models: the even rank-4 hexadecapole and the odd-rank-5 dotriacontapole model. We obtain the quasiparticle dispersion and reconstructed Fermi surface in each HO phase adapting an effective two-orbital model of 5f bands that reproduces the main Fermi surface sheets of the para phase. We show that the resulting QPI spectrum reflects directly the effect of fourfold symmetry breaking in the rank-5 model which is absent in the rank-4 model. Therefore we suggest that QPI method should give a possibility of direct discrimination between the two most investigated models of HO in URu2Si2. Furthermore the signature of proposed chiral d-wave superconducting (SC) order parameter in QPI of the coexisting HO+SC phase is investigated.

Systematic Improvement of Potential-Derived Atomic Multipoles and Redundancy of the Electrostatic Parameter Space.

We assess the accuracy of force field (FF) electrostatics at several levels of approximation from the standard model using fixed partial charges to conformational specific multipole fits including up to quadrupole moments. Potential-derived point charges and multipoles are calculated using least-squares methods for a total of ∼1000 different conformations of the 20 natural amino acids. Opposed to standard charge fitting schemes the procedure presented in the current work employs fitting points placed on a single isodensity surface, since the electrostatic potential (ESP) on such a surface determines the ESP at all points outside this surface. We find that the effect of multipoles beyond partial atomic charges is of the same magnitude as the effect due to neglecting conformational dependency (i.e., polarizability), suggesting that the two effects should be included at the same level in FF development. The redundancy at both the partial charge and multipole levels of approximation is quantified. We present an algorithm which stepwise reduces or increases the dimensionality of the charge or multipole parameter space and provides an upper limit of the ESP error that can be obtained at a given truncation level. Thereby, we can identify a reduced set of multipole moments corresponding to ∼40% of the total number of multipoles. This subset of parameters provides a significant improvement in the representation of the ESP compared to the simple point charge model and close to the accuracy obtained using the complete multipole parameter space. The selection of the ∼40% most important multipole sites is highly transferable among different conformations, and we find that quadrupoles are of high importance for atoms involved in π-bonding, since the anisotropic electric field generated in such regions requires a large degree of flexibility.

Homogenization scheme for acoustic metamaterials

We present a homogenization scheme for acoustic metamaterials that is based on reproducing the lowest orders of scattering amplitudes from a finite volume of metamaterials. This approach is noted to differ significantly from that of coherent potential approximation, which is based on adjusting the effective-medium parameters to minimizescatteringsinthelong-wavelengthlimit.Withtheaidofmetamaterials’eigenstates,theeffectiveparameters, such as mass density and elastic modulus can be obtained by matching the surface responses of a metamaterial’s structuralunitcellwithapieceofhomogenizedmaterial.FromtheGreen’stheoremappliedtotheexteriordomain problem, matching the surface responses is noted to be the same as reproducing the scattering amplitudes. We verifyourschemebyapplyingittothreedifferentexamples:alayeredlattice,atwo-dimensionalhexagonallattice, and a decorated-membrane system. It is shown that the predicted characteristics and wave fields agree almost exactly with numerical simulations and experiments and the scheme’s validity is constrained by the number of dominant surface multipoles instead of the usual long-wavelength assumption. In particular, the validity extends to the full band in one dimension and to regimes near the boundaries of the Brillouin zone in two dimensions.

Structure of two-dimensional plasma crystals in anharmonic Penning traps

This paper derives an analytic expression for the density per unit area of charges confined in a two-dimensional configuration in a Penning trap with an anharmonic applied trap potential that is expressed as a multipole expansion. This expression is used to find the optimum potential, with a given number of multipoles, for trapping a plasma with the most uniform possible density per unit area. Minimum energy states in such an optimized trap potential are evaluated numerically and the resulting crystal structures are shown to be defect-free over the central region of the plasma where the density is most nearly uniform. The paper also briefly considers the possibility of using an $\ensuremath{\ell}=3$ rotating wall trap potential in order to confine minimum energy states with triangular symmetry and no defects.

Light scattering by a cube: Accuracy limits of the discrete dipole approximation and the T-matrix method

We simulated light-scattering by small and wavelength-sized cubes with three largely different values of the refractive index using the discrete dipole approximation (DDA) and the T-matrix method. Our main goal was to push the accuracy of both methods to the limit. For the DDA we used an earlier developed extrapolation technique based on simulation results for different levels of discretization. For the T-matrix method we developed a procedure to estimate a confidence range for the simulated value, using results for different values of the truncation index (number of multipoles). In most cases this confidence range was reliable, enclosing the corresponding DDA result. We present benchmark results by both methods, including estimated uncertainties, for selected integral and angle-resolved scattering quantities. Estimated relative uncertainties of the DDA result are unprecedentedly small (from 10−7 to 10−3), while relative differences between the T-matrix and DDA results are larger (from 10−4 to 0.2) in accordance with estimated T-matrix uncertainties.

Efficient Implementation of Inner-Outer Flexible GMRES for the Method of Moments Based on a Volume-Surface Integral Equation

This paper presents flexible inner-outer Krylov subspace methods, which are implemented using the fast multipole method (FMM) for solving scattering problems with mixed dielectric and conducting object. The flexible Krylov subspace methods refer to a class of methods that accept variable preconditioning. To obtain the maximum efficiency of the inner-outer methods, it is desirable to compute the inner iterations with the least possible effort. Hence, generally, inaccurate matrix-vector multiplication (MVM) is performed in the inner solver within a short computation time. This is realized by using a particular feature of the multipole techniques. The accuracy and computational cost of the FMM can be controlled by appropriately selecting the truncation number, which indicates the number of multipoles used to express far-field interactions. On the basis of the abovementioned fact, we construct a less-accurate but much cheaper version of the FMM by intentionally setting the truncation number to a sufficiently low value, and then use it for the computation of inaccurate MVM in the inner solver. However, there exists no definite rule for determining the suitable level of accuracy for the FMM within the inner solver. The main focus of this study is to clarify the relationship between the overall efficiency of the flexible inner-outer Krylov solver and the accuracy of the FMM within the inner solver. Numerical experiments reveal that there exits an optimal accuracy level for the FMM within the inner solver, and that a moderately accurate FMM operator serves as the optimal preconditioner.

Quantitative Evaluation for Computational Cost of CG-FMM on Typical Wiregrid Models

The conjugate gradient-fast multipole method (CG-FMM) is one of the powerful methods for analysis of large-scale electromagnetic problems. It is also known that CPU time and computer memory can be reduced by CG-FMM but such computational cost of CG-FMM depends on shape and electrical properties of an analysis model. In this paper, relation between the number of multipoles and number of segments in each group is derived from dimension of segment arrangement in four typical wiregrid models. Based on the relation and numerical results for these typical models, the CPU time per iteration and computer memory are quantitatively discussed. In addition, the number of iteration steps, which is related to condition number of impedance matrix and analysis model, is also considered from a physical point of view.

Electrostatics in the presence of spherical dielectric discontinuities

A united description of the electrostatics of an arbitrary number of electrostatic multipoles, each localized in a spherical dielectric cavity, in a dielectric medium is presented. The permanent charges as well as the polarization surface charges are described by multipole expansions in standard format. Expressions of the polarization surface charge density, the electrostatic potential energy, and the electrostatic interaction including the contribution from the polarization surface charge densities are given. Interacting electrostatic multipoles in dielectric spheres immersed in a medium with a higher (lower) relative dielectric permittivity experience a repulsive (attractive) potential term that increases in magnitude at reduced multipole separation, originating from the polarization surface charges appearing at the dielectric interfaces. Simplified expressions applied to monopoles and to two dielectric cavities are provided. Numerical examples involving monopoles and dipoles quantifying the effect of the surface polarization are also included.

On the accuracy of multipole expansion of deformed density distribution

The interaction potential for a deformed-spherical pair of nuclei is calculated using the folding model derived from different range nucleon–nucleon (NN) interactions. Five spherical projectiles of different mass numbers scattered on the 238U deformed target are considered. The error in the heavy ion (HI) potential by using the truncated multipole density expansion is evaluated for each case. We find systematic trends of the percentage error in the HI potential depending on the number of multipoles considered; this percentage decreases if the mass number of the projectile or the range of the NN force increases, and it becomes smaller for small values of the separation distance between two nuclei or when higher deformation parameters vanish. The maximum error in using the truncated density expansion is estimated in the case of two deformed interacting nuclei.

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.

Multilevel multipole and local operators for potentials of the form [formula omitted

This paper presents multilevel multipole and local operators for potentials of the form r − λ . A new addition theorem which enables the construction of the multilevel operators is stated and proved. Simulations of the new operators are presented that indicate the reduction of error with increasing number of multipoles and O ( N ) complexity of the method.