Vector Spherical Harmonics Research Articles

Estimation of large-scale deformations in VLBI radio source catalogs with mitigation of impact of outliers: A comparison between different L1- and L2-norm-based methods

Aims. In this study, we compare several methods of modeling large-scale systematic differences between catalogs of positions of extragalactic radio sources provided by very long baseline interferometry with an emphasis on mitigating the impact of outliers. Methods. The coordinate difference between catalogs was parameterized by first and second order coefficients of vector spherical harmonics. We solved for these coefficients by using the least-squares method (L2-norm) and, alternatively, by L1-norm minimization. The problem of outliers was addressed either by rejecting them on the basis of their difference to the mean or by using the cell median (CM) method, consisting of reducing the difference field to median values in cells of equal area. The methods were applied to simulated catalogs exhibiting systematics of within 100 microarcseconds - for which we knew the expected results - and to real catalogs. Results. In simulations, the L1 minimization appears practically insensitive to outliers and is within a few microarcseconds of the expected results. Least-squared fitting preceded by L1-norm-based outlier detection performed similarly. The CM method gets close to the true parameter values, within one microarcsecond. When applied to real catalogs, all methods provide close results within a few microarcseconds. Conclusions. The study shows that all tested methods are consistent with each other within a few microarcseconds. Hybrid L2/L1 and iterative L2 methods proved to be very effective in eliminating outliers and showed the best accuracy of the estimated parameters of the mutual orientation of celestial reference frames.

Normal modes of Proca fields in AdS spacetime

A normal mode analysis for Proca fields in the anti-de Sitter (AdS) spacetime is given. It is found that the equations for the Proca field can be decoupled analytically. This is performed by changing the basis of the vector spherical harmonics decomposition. The normal modes and the normal mode frequencies of the Proca equation in the AdS spacetime are then analytically determined. It is also shown that the Maxwell field can be recovered by taking the massless limit of the Proca field with care so that the nonphysical gauge modes are eliminated.

THE HIPPARCOS PLEIADES PARALLAX ERROR IS ALSO A PROPER MOTION ERROR

The mean parallax of the Pleiades open cluster from the Hipparcos catalog is larger than the true value by approximately 1 mas. The origin of this error, as well as a possible algorithm of correcting it, was proposed by Makarov (2002). The problem is reassessed using the more accurate Gaia data with a focus on the predicted correction to the Pleiades proper motions. The accurately determined differences Gaia − Hipparcos for 52 common stars are close to these estimates within the formal uncertainties for all three parameters, which strongly suggests that the proposed interpretation was correct. With adjustments for the systematic vector field fitted with 126 vector spherical harmonics to degree 7, these differences amount to (+0.39, −0.74) mas yr−1 . The implications of small-scale proper motion and position errors in Hipparcos for present day astrometry are briefly discussed.

Differential Proper Motion Spin of the Hipparcos and Gaia Celestial Reference Frames

The Hipparcos catalog provides the first epoch of the celestial reference frame (CRF) in the optical domain and serves as an indispensable tool to verify and improve the Gaia CRF for the brighter stars (V < 11 mag) and to identify the elusive astrometric binary stars with dim or invisible companions, including long-period exoplanets. The systems of positions in Hipparcos and Gaia cannot be directly compared, because they refer to two different mean epochs. It is shown that the proper motion systems for carefully filtered common stars are not statistically consistent within the given formal errors. The vector field of proper motion differences is fitted with 126 vector spherical harmonics up to degree 7 revealing a global pattern at high signal-to-noise ratios, including the three terms of rigid rotation. The origin of the differential spin and other large harmonic terms is investigated by producing a similar decomposition of the Gaia−HG proper motion field, where HG stands for the long-term proper motions derived from the Hipparcos and Gaia DR3 positions, for the same sample of stars. Hipparcos proper motions emerge as the largest source of sky-correlated distortions of the multiepoch optical CRF with a median value of ∼190 μas yr−1 and a global spin of ∼226 μas yr−1, while the Hipparcos positions and Gaia EDR3 proper motions are explicitly consistent by construction at a level of ∼10 μas yr−1. The latter, however, include multiple distortions of higher degree, which should be taken into account in astrometric applications using the HG field.

Exact solutions of time-dependent oscillations in multipolar spherical vortices

Exact solutions of the time-dependent three-dimensional nonlinear vorticity equation for Euler flows with spherical geometry are provided. The velocity solution is the sum of a multipolar oscillatory function and a rigid cylindrical motion with swirl. The multipolar oscillation is a velocity mode whose radial and angular dependencies are given by the spherical Bessel functions and vector spherical harmonics, respectively. The local frequency of the velocity oscillations equals the angular speed of the rigid flow times the angular azimuthal wavenumber of the oscillating flow. The unsteady motion corresponds to inertial oscillations in multipolar flows with spatial azimuthal waves (non-vanishing azimuthal wavenumber) in the presence of a background flow with constant axial vorticity. In these nonlinear solutions, the curl of the Lamb vector has a linear dependence with the oscillation velocity, a property that makes it possible for the oscillating motion to satisfy different linear wave equations. Based on these inviscid time-dependent velocity modes, new exact solutions to the time-dependent Navier–Stokes equation are also provided.

Lamb-type solution and properties of unsteady Stokes equations

We derive the general solution of the unsteady Stokes equations in unbounded viscous fluid exterior to a sphere. The solution is an expansion in vector spherical harmonics and it uses the decomposition originally introduced by Lamb. The coefficients of the expansion are uniquely determined by the arbitrary velocity distribution at the spherical boundary. The proposed solution can be applied, for example, to construct transient flow around a spherical ``squirmer'', to study particle-particle and particle-wall interactions in oscillatory flows and in other applications.

Vector spherical wave function truncation in the invariant imbedding T-matrix method.

Both the computational costs and the accuracy of the invariant-imbedding T-matrix method escalate with increasing the truncation number N at which the expansions of the electromagnetic fields in terms of vector spherical harmonics are truncated. Thus, it becomes important in calculation of the single-scattering optical properties to choose N just large enough to satisfy an appropriate convergence criterion; this N we call the optimal truncation number. We present a new convergence criterion that is based on the scattering phase function rather than on the scattering cross section. For a selection of homogeneous particles that have been used in previous single-scattering studies, we consider how the optimal N may be related to the size parameter, the index of refraction, and particle shape. We investigate a functional form for this relation that generalizes previous formulae involving only size parameter, a form that shows some success in summarizing our computational results. Our results indicate clearly the sensitivity of optimal truncation number to the index of refraction, as well as the difficulty of cleanly separating this dependence from the dependence on particle shape.

Vector Spherical Harmonics for Data‐Assimilative Neutral Wind Estimation

AbstractIn this work we enable the data assimilation algorithm Estimating Model Parameters from Ionospheric Reverse Engineering (EMPIRE) to estimate global neutral winds. EMPIRE corrects the ion drifts and neutral winds from background models using electron density in the F region derived mainly from total electron content measurements. The new EMPIRE basis functions for the neutral winds are vector spherical harmonics, enforcing the field to be smooth and continuous globally. Global basis functions allow us to estimate the horizontal wind vector from time‐varying plasma densities. The geomagnetic storm on 25 October 2011 is studied to investigate the global implementation of neutral wind estimation. During this storm, the estimates are also compared to those of the old method of estimation using power series and to Fabry‐Perot interferometer (FPI) measurements for validation at three different sites: Pisgah, Cariri, and Nasca. The new global estimation is 10%–20% closer to the FPI measurements than the model and than the old method for most of the viewing directions of the sites, in terms of root‐mean‐square residuals. In the northward direction at Pisgah the estimates using either the old or new implementation disagree with the model, and might be related to an insufficiently small temporal scale used in EMPIRE.

Vector spherical harmonic analysis and experimental validation of spherical shells illuminated with broadband, millimeter wave Gaussian beams: applications to corneal sensing.

Coupling to longitudinal modes of thin spherical shells, under Gaussian-beam illumination, was explored with a theoretical method based on Fourier-optics analysis and vector spherical harmonics and was scrutinized with an experimental setup. For the theory part, the illumination frequency band was fixed between 100-600 GHz and the outer spherical shell radius of curvature and thickness are 7.5 mm and 0.5 mm, respectively. The shell material was either the lossless cornea or an aqueous effective media representing the cornea. Six different beam-target strategies were introduced being potential candidates for maximum coupling. Two dispersion-tuned beam ensembles with strongly frequency-dependent phase center location have been created with a fixed incident beam 1/e radius and radius of curvature called forward strategies. These computations of different alignments were continued with four beam ensembles of frequency-invariant phase center, constructed from fits to experimental data, oriented at four different axial locations with respect to the spherical shell center of curvature, they are called reverse strategies. Coupling efficiency for all strategies was calculated for different targets including perfect electrical conductor (PEC) sphere, PEC core covered by a cornea loss-free layer and cornea. All scattering strategies contrasted to scattering from equivalent planar targets as a reference with maximum coupling. The results show that, under an ideal calibration, forward strategies are a closer approximation to the plane-wave condition for the cornea. An experimental setup was assembled to explore the simulation approach in a frequency range between 220 GHz to 330 GHz. Two different quartz samples with permittivity of 4.1 were mounted on a water core, acting for a cornea. The first and second quartz radius and thickness were 7.5 mm and 0.5 mm and 8 mm and 1 mm, respectively. An adequate agreement between theory and experiment was confirmed. A particle optimisation swarm algorithm was applied to extract the thickness and permittivity of quartz from the measured back-scattered field for reverse strategies.

Spherical harmonic based noise rejection and neuronal sampling with multi-axis OPMs

In this study we explore the interference rejection and spatial sampling properties of multi-axis Optically Pumped Magnetometer (OPM) data. We use both vector spherical harmonics and eigenspectra to quantify how well an array can separate neuronal signal from environmental interference while adequately sampling the entire cortex. We found that triaxial OPMs have superb noise rejection properties allowing for very high orders of interference (L=6) to be accounted for while minimally affecting the neural space (2dB attenuation for a 60-sensor triaxial system). We show that at least 11th order (143 spatial degrees of freedom) irregular solid harmonics or 95 eigenvectors of the lead field are needed to model the neural space for OPM data (regardless of number of axes measured). This can be adequately sampled with 75–100 equidistant triaxial sensors (225–300 channels) or 200 equidistant radial channels. In other words, ordering the same number of channels in triaxial (rather than purely radial) configuration may give significant advantages not only in terms of external noise rejection but also by minimizing cost, weight and cross-talk.

Scattering of elastic waves by a transversely isotropic sphere and ultrasonic attenuation in hexagonal polycrystalline materials

The scattering of elastic waves by a transversely isotropic sphere in an isotropic medium is considered. The elastodynamic equations inside the sphere are transformed to spherical coordinates and the displacement field is expanded in the vector spherical harmonics in the angular coordinates and powers in the radial coordinate. The governing equations inside the sphere then give recurrence relations among the expansion coefficients. Then all the remaining expansion coefficients for the fields outside and inside the sphere are found using the boundary conditions on the surface of the sphere. As a result, the transition (T) matrix elements are calculated and given explicitly for low frequencies. Using the T matrix and the theory of Foldy an explicit expression for the effective complex wave number of transversely isotropic (hexagonal) polycrystalline materials are presented for low frequencies. Numerical comparisons are made with previously published results and with recent FEM results and show a very good correspondence with FEM for low frequencies. As opposed to other published methods there is no limitation on the degree of anisotropy with the present approach.

Terahertz bound state in the continuum in dielectric membrane metasurfaces

Mie-resonant metasurfaces composed of subwavelength dielectric resonators enable an efficient route for electromagnetic wave manipulation. Among these manipulations, a localized mode with a high-quality factor coexisting with a continuous spectrum of radiating waves termed bound state in the continuum (BIC) can arouse many exotic applications in photonics. Here, we demonstrate the terahertz BIC in a dielectric membrane metasurface and analyze its resonant nature based on Mie-resonant multipoles and vector spherical harmonics. The intrinsic splitting of the resonances under oblique incidence is also explored, in which the conversion of multipole radiation patterns versus the oblique angle will drive the resonances from BIC to leaky modes or vice versa. Both Γ and off-Γ point BICs could be identified as the superposition cancellation of vector spherical harmonics for both p-wave and s-wave. Our research not only provides a novel perspective for exploring the essence of BIC metasurfaces in the terahertz regime, but also points new opportunities for achieving terahertz BIC metasurfaces with ultra-high quality factors.

Optimal Sampling in Spherical Near-Field Antenna Measurements by Utilizing the Information Content of Spherical Wave Harmonics

In this article, we propose an efficient method that specifies the minimum number of samples and their locations to measure the spherical near-field (SNF) of antennas utilizing information-theoretic tools. To this end, spherical harmonics are investigated from the information-theoretic point of view. Our study determines the minimum required samples and provides a validation tool to assess the ability of a sampling scheme to recover the antenna characteristics by the measured data. Furthermore, an optimum sampling scheme is proposed by clustering the energy of vector spherical harmonics using the frame theory without any time-consuming iterative, which leads to a highly efficient SNF measurement procedure. The number of samples determined by the proposed method is only 50% of the Nyquist method, which provides significant advantages for storage and processing time. Our method additionally shows a great promise for the recovery of the far-field radiation pattern from near-field samples with the same performance as the Nyquist sampling on a sphere. To derive the proposed sampling scheme with the minimum redundancy, no prior information on the antenna is assumed. Finally, the proposed sampling scheme is validated on measurement data.

Calibration and Localization of Optically Pumped Magnetometers Using Electromagnetic Coils.

In this paper, we propose a method to estimate the position, orientation, and gain of a magnetic field sensor using a set of (large) electromagnetic coils. We apply the method for calibrating an array of optically pumped magnetometers (OPMs) for magnetoencephalography (MEG). We first measure the magnetic fields of the coils at multiple known positions using a well-calibrated triaxial magnetometer, and model these discreetly sampled fields using vector spherical harmonics (VSH) functions. We then localize and calibrate an OPM by minimizing the sum of squared errors between the model signals and the OPM responses to the coil fields. We show that by using homogeneous and first-order gradient fields, the OPM sensor parameters (gain, position, and orientation) can be obtained from a set of linear equations with pseudo-inverses of two matrices. The currents that should be applied to the coils for approximating these low-order field components can be determined based on the VSH models. Computationally simple initial estimates of the OPM sensor parameters follow. As a first test of the method, we placed a fluxgate magnetometer at multiple positions and estimated the RMS position, orientation, and gain errors of the method to be 1.0 mm, 0.2°, and 0.8%, respectively. Lastly, we calibrated a 48-channel OPM array. The accuracy of the OPM calibration was tested by using the OPM array to localize magnetic dipoles in a phantom, which resulted in an average dipole position error of 3.3 mm. The results demonstrate the feasibility of using electromagnetic coils to calibrate and localize OPMs for MEG.

Local Multiple Traces Formulation for electromagnetics: Stability and preconditioning for smooth geometries

We consider the time-harmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple traces formulation (MTF) for electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes (2012) for the acoustic case, paving the way towards an extension to general piecewise homogeneous scatterers. Moreover, we investigate preconditioning techniques for the local MTF scheme and study the accumulation points of induced operators. In particular, we propose a novel second-order inverse approximation of the operator. Numerical experiments validate our claims and confirm the relevance of the preconditioning strategies proposed.

Numerical evaluation of time-distance helioseismic sensitivity kernels in spherical geometry

Context. Helioseismic analysis of large-scale flows and structural inhomogeneities in the Sun requires the computation of sensitivity kernels that account for the spherical geometry of the Sun, as well as systematic effects such as line-of-sight projection. Aims. I aim to develop a code to evaluate helioseismic sensitivity kernels for flows using line-of-sight projected measurements. Methods. I decomposed the velocity field in a basis of vector spherical harmonics and computed the kernel components corresponding to the coefficients of velocity in this basis. The kernels thus computed are radial functions that set up a 1.5D inverse problem to infer the flow from surface measurements. I demonstrate that using the angular momentum addition formalism lets us express the angular dependence of the kernels as bipolar spherical harmonics, which may be evaluated accurately and efficiently. Results. Kernels for line-of-sight projected measurements may differ significantly from those that don’t account for projection. Including projection in our analysis does not increase the computational time significantly. We demonstrate that it is possible to evaluate kernels for pairs of points that are related through a rotation by linearly transforming the terms that enter the expression of the kernel, and that this result holds even for line-of-sight projected kernels. Conclusions. I developed a Julia code that may be used to evaluate sensitivity kernels for seismic wave travel times computed using line-of-sight projected measurements, which is made freely available under the MIT license.

Quasinormal modes of Proca fields in a Schwarzschild-AdS spacetime

We present new results concerning the Proca massive vector field in a Schwarzschild-AdS black hole geometry. We provide a first principles analysis of Proca vector fields in this geometry using both the vector spherical harmonic (VSH) separation method and the Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k-Santos (FKKS) method that separates the relevant equations in spinning geometries. The analysis in the VSH method shows, on one hand, that it is arduous to separate the scalar-type from the vector-type polarizations of the electric sector of the Proca field, and on the other hand, it displays clearly the electric and the magnetic mode sectors. The analysis in the FKKS method is performed by taking the nonrotating limit of the Kerr-AdS spacetime, and shows that the ansatz decouples the polarizations in the electric mode sector even in the nonrotating limit. On the other hand, it captures only two of the three possible polarizations, the magnetic mode sector is missing. The reason for the absence of this polarization is related to the degeneracy of the principal tensor in static spherical symmetric spacetimes. The degrees of freedom and quasinormal modes in both separation methods of the Proca field are found. The frequencies of the quasinormal modes are also computed. For the electric mode sector in the VSH method the frequencies are found through an extension, which substitutes number coefficients by matrix coefficients, of the Horowitz-Hubeny numerical procedure, whereas for the magnetic mode sector in the VSH method and the electric sector of the FKKS method it is shown that a direct use of the procedure can be made. The values of the quasinormal mode frequencies obtained for each method are compared and showed to be in good agreement with each other. This further supports the analytical approaches presented here for the behavior of the Proca field in a Schwarzschild-AdS black hole background.

Multipolar spherical and cylindrical vortices

Multipolar spherical solutions to the three-dimensional steady vorticity equation are provided. These solutions are based on the separation of radial and angular contributions in terms of the spherical Bessel functions and vector spherical harmonics, respectively. In this set of multipolar vortex solutions, the Hicks–Moffatt swirling vortex is categorized as a vortex of degree ${\ell }=1$ and therefore as a vortex dipole. This swirling vortex is the three-dimensional dipole in spherical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry. The three-dimensional dipole solution admits two linearly superposable solutions. The first one is a Trkalian flow and the second one is a cylindrical solid-body rotation with swirl. The higher ${\ell }&gt;1$ multipolar vortices found are either vanishing-helicity vortices or Trkalian flow vortices. The multipolar Trkalian flows admit two circular polarizations given by the sign of the wavenumber $k$ . It is also found that piecewise vortex solutions, consisting of interior rotational and exterior potential flow domains, satisfying velocity continuity conditions at the vortex boundary, are possible in the general multipolar Trkalian spherical vortex. A particular polarized dipole solution in three-dimensional cylindrical geometry, consisting as well in the superposition of a Trkalian flow and a rigid motion, is also analysed. This swirling vortex may be interpreted as the three-dimensional dipole in cylindrical geometry equivalent to the two-dimensional Lamb–Chaplygin dipole in polar geometry.

Application of Vector Spherical Harmonics to the Magnetization of Mars' Crust

AbstractMars has a magnetic field originating in its strongly magnetized crust that holds clues to the planet's interior. We apply vector spherical harmonic decomposition to simple candidate magnetic structures to separate the parts responsible for the anomalies from those that remain invisible. A uniform magnetic layer produces no anomalies: spatial variations are essential although secondary magnetization does produce a weak field that might reflect the primordial dynamo field. A hemispheric layer produces anomalies confined to the equator rather than the observed hemispheric difference. A uniformly magnetized crust with variable thickness determined from gravity and topography produces a crustal field with large anomalies at the major impact crater sites that are not observed. These anomalies are not present if the magnetic layer lies deeper than the crater floor. We conclude that decomposing magnetizations in this way is a useful tool in the interpretation of Martian magnetic anomalies.

Modeling of luminescence spectra in spherical microresonators with an emitting shell

The luminescence spectra of a microresonator structure consisting of a spherical core of small diameter (3.5-6 mcm) covered with a luminescent shell with a refractive index less than that of the core are modeled. Shell luminescence spectra, radial distribution of the whispering gallery mode (WGM) field, and mode parameters (wavelength, width, quality factor) are calculated using the expansion of the electromagnetic wave field in the basis of vector spherical harmonics and the method of spherical wave transfer matrices. The dependence of the luminescence spectra and WGM parameters on the geometric and optical parameters of the structure --- the shell thickness, the refractive index of the shell, and the core diameter --- is studied. Keywords: spherical microresonator, luminescent shell, whispering gallery modes, modeling of luminescence spectra.