Vector Spherical Harmonic Expansion Research Articles

VSHPIC: a particle-in-cell algorithm based on vector spherical harmonics expansion

The particle-in-cell (PIC) method has been widely used for studying plasma physics. However, fully three-dimensional PIC simulations always require huge computational resources. For problems with near azimuthal symmetry, recent work (Lifschitz et al 2009 J. Comput. Phys. 228 1803–14, Davidson et al 2015 J. Comput. Phys. 281 1063–77, Li et al 2021 Comput. Phys. Commun. 261 107784, Li et al 2022 J. Comput. Phys. 470 111599) has shown that expanding all the quantities defined on the grid in azimuthal harmonics and truncating the expansion can improve the code efficiency. In this paper, we describe a novel parallel algorithm for efficiently simulating three-dimensional near-spherical symmetry problems. Our approach expands all physical quantities in the θ and ϕ directions in spherical coordinates using vector spherical harmonics. The code is capable of simulating three-dimensional asymmetric scenarios by accurately tracking the evolution of distinct individual modes while preserving the charge conservation law. The fundamental dispersion relation of EM waves in the plasma has been obtained using VSHPIC simulation results. The code also shows a well strong scalability up to more than 1000 cores.

How Well is the International Celestial Reference System Maintained in Official IAU Implementations?

The International Celestial Reference Frame (ICRF) based on the VLBI-derived positions of 608 extragalactic radio sources was adopted by the International Astronomical Union (IAU) in 1998 as the first realization of the International Celestial Reference System (ICRS). Later, in 2009 and 2020, two extended ICRF versions, ICRF2 and ICRF3, respectively, were released. The latter is adopted by the IAU as the current implementation of the ICRF in the radio band. In the meantime, the Gaia mission delivered three versions of the optical ICRS realization in 2016, 2018, and 2022 with an accuracy similar to that achieved by VLBI-based ICRF. The Gaia-CRF catalogs were linked to the ICRF under no-rotation conditions and thus may suffer from ICRF systematic instability if the latter is substantial. In this work, a new analysis was performed to assess the long-term stability of radio and optical ICRS realizations. Based on the 16-parameter vector spherical harmonics expansion of the differences between the three ICRF catalogs, it can be concluded that the mutual orientation between them is at a level of a few tens of microarcseconds, while the components of the glide vector and E 2,0 term are several times greater. A comparison of the three Gaia-CRF catalogs with the ICRF3-SX showed that for the latest Gaia-CRF catalog, Gaia-CRF3, all rotational and deformation components are below 20 μas except for the E 2,0 term, which is several times greater. For both ICRF and Gaia-CRF catalogs, the evolution of the source position errors is also tracked.

Mutual impedance computation using spherical harmonics between two closely spaced antennas

Mutual coupling between two identical radiators is computed using vector spherical harmonics and reaction theorem for small element spacing cases. For closely spaced antennas, the minimum spheres that enclose the antennas overlap, and the vector spherical harmonic expansions do not converge on the surface that separates the two antennas. The proposed method mainly uses addition theorems to translate the vector spherical harmonic expansions to a different center. These operations on spherical waves enable to redefine the minimum spheres and convergence regions. The proposed method has been validated by analyzing the coupling between two half-wave dipoles in a side-by-side configuration. The results clearly show excellent agreement with that obtained from induced EMF method. The theory is also tested experimentally. A printed dipole antenna is designed, and its spherical harmonic coefficients are obtained by spherical near field measurement. Mutual impedance between two such dipoles is computed using the proposed method. The computed results for different element spacings in a side-by-side configuration are compared to those obtained from s-parameter measurements.

Towards analytical calculation of the magnetic flux measured by magnetometers

This paper reviews magnetic flux signal calculations through pick-up loops using vector spherical harmonic expansion under the quasi-static approximation, and presents a near-analytical method of evaluating the flux through arbitrary parametrizable pick-up loops for each expansion degree. This is done by simplifying the surface flux integral (2D) into a line integral (1D). For the special case of tangential circular sensors, we present a fully analytical recursion calculation. We then compare commonly-used cubature approximations to our (near-)analytical forms, and show that cubature approximations suffer from increasing errors for higher spatial frequency components. This suggests the need for more accurate evaluations for increasingly sensitive sensors that are being developed, and our (near-)analytical forms themselves are a solution to this problem.

Spherical harmonics expansion of the evanescent waves in angular spectrum decomposition of shaped beams

The angular spectrum decomposition (ASD) is a powerful tool for analyzing the interaction between spherical particles and shaped beams, in which homogeneous plane waves and/or evanescent waves are expanded into series of vector spherical harmonics (VSH). The paper presents a numerical study on the VSH expansion of the evanescent wave and its behavior in the field reconstruction of the shaped beam. It is revealed that the beam shape coefficients (BSCs) of the evanescent waves increase exponentially along the increasing partial wave number, consequently extending the convergence of the series and restricting the field reconstruction into a limited space.

Generating geometric body shapes with electromagnetic source scattering techniques

<p style='text-indent:20px;'>In this paper, we are concerned with the three-dimensional (3D) geometric body shape generation with several well-selected characteristic values. Since 3D human shapes can be viewed as the support of the electromagnetic sources, we formulate a scheme to regenerate 3D human shapes by inverse scattering theory. With the help of vector spherical harmonics expansion of the magnetic far field pattern, we build on a smart one-to-one correspondence between the geometric body space and the multi-dimensional vector space that consists of all coefficients of the spherical vector wave function expansion of the magnetic far field pattern. Therefore, these coefficients can serve as the shape generator. For a collection of geometric body shapes, we obtain the inputs (characteristic values of the body shapes) and the outputs (the coefficients of the spherical vector wave function expansion of the corresponding magnetic far field patterns). Then, for any unknown body shape with the given characteristic set, we use the multivariate Lagrange interpolation to get the shape generator of this new shape. Finally, we get the reconstruction of this unknown shape by using the multiple-frequency Fourier method. Numerical examples of both whole body shapes and human head shapes verify the effectiveness of the proposed method.</p>

Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids

We present in this paper a spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map.

Estimating network effect in geocenter motion: Theory

AbstractGeophysical models and their interpretations of several processes of interest, such as sea level rise, postseismic relaxation, and glacial isostatic adjustment, are intertwined with the need to realize the International Terrestrial Reference Frame. However, this realization needs to take into account the geocenter motion, that is, the motion of the center of figure of the Earth surface, due to, for example, deformation of the surface by earthquakes or hydrological loading effects. Usually, there is also a discrepancy, known as the network effect, between the theoretically convenient center of figure and the physically accessible center of network frames, because of unavoidable factors such as uneven station distribution, lack of stations in the oceans, disparity in the coverage between the two hemispheres, and the existence of tectonically deforming zones. Here we develop a method to estimate the magnitude of the network effect, that is, the error introduced by the incomplete sampling of the Earth surface, in measuring the geocenter motion, for a network of space geodetic stations of a fixed size N. For this purpose, we use, as our proposed estimate, the standard deviations of the changes in Helmert parameters measured by a random network of the same size N. We show that our estimate scales as and give an explicit formula for it in terms of the vector spherical harmonics expansion of the displacement field. In a complementary paper we apply this formalism to coseismic displacements and elastic deformations due to surface water movements.

An Efficient Spectral-Galerkin Approximation and Error Analysis for Maxwell Transmission Eigenvalue Problems in Spherical Geometries

We propose and analyze an efficient spectral-Galerkin approximation for the Maxwell transmission eigenvalue problem in spherical geometry. Using a vector spherical harmonic expansion, we reduce the problem to a sequence of equivalent one-dimensional TE and TM modes that can be solved individually in parallel. For the TE mode, we derive associated generalized eigenvalue problems and corresponding pole conditions. Then we introduce weighted Sobolev spaces based on the pole condition and prove error estimates for the generalized eigenvalue problem. The TM mode is a coupled system with four unknown functions, which is challenging for numerical calculation. To handle it, we design an effective algorithm using Legendre-type vector basis functions. Finally, we provide some numerical experiments to validate our theoretical results and demonstrate the efficiency of the algorithms.

General-relativistic electromagnetic fields around a slowly rotating neutron star: stationary vacuum solutions

Pulsars are thought to be highly magnetized rotating neutron stars accelerating charged particles along magnetic field lines in their magnetosphere and visible as pulsed emission from the radio wavelength up to high energy X-rays and gamma-rays. Being highly compact objects with compactness close to $\Xi = R_s/R\approx0.5$, where $\Rs=2\,G\,M/c^2$ is the Schwarzschild radius and $\{M,R\}$ the mass and radius of the neutron star, general-relativistic effects become important close to their surface. This is especially true for the polar caps where radio emission is supposed to emanate from, leading to well defined signatures such as linear and circular polarization. In this paper, we derive a general formalism to extend to general relativity the Deutsch field solution valid in vacuum space. Thanks to a vector spherical harmonic expansion of the electromagnetic field, we are able to express the solution to any order in the spin parameter $\Omega$ of the compact object. We hope this analysis to serve as a benchmark to test numerical codes used to compute black hole and neutron star magnetospheres.

Ultra-compact parametric modelling of three-dimensional antenna radiation pattern for impulse radio-ultra-wide band ray tracing

In ultra-wide band (UWB) propagation channel modelling, antenna effects – namely frequency and polarisation aspects – must be considered to provide realistic waveform characteristics and realistic signal received power. The authors propose an efficient method which enables ultra-compact parametric modelling of fully three-dimensional antenna radiation behaviour for UWB ray tracing channel simulation tools. Two complementary techniques based on a vector spherical harmonic (VSH) expansion of antenna radiation function are proposed as a solution for (i) compression of the amount of data needed for describing antenna characteristics and (ii) fast and reliable reconstruction of antenna array response for waves coming from arbitrary directions and with arbitrary frequency variations. The most significant VSH coefficients are first extracted and then compressed in the frequency domain. The method is evaluated through the study of an UWB monocone antenna. It is shown that data compression rates as high as 1:1719 are achievable in practice.

The pulsar force-free magnetosphere linked to its striped wind: time-dependent pseudo-spectral simulations

(abridged) Pulsar activity and its related radiation mechanism are usually explained by invoking some plasma processes occurring inside the magnetosphere. Despite many detailed local investigations, the global electrodynamics around those neutron stars remains poorly described. Better understanding of these compact objects requires a deep and accurate knowledge of their immediate electromagnetic surrounding within the magnetosphere and its link to the relativistic pulsar wind. The aim of this work is to present accurate solutions to the nearly stationary force-free pulsar magnetosphere and its link to the striped wind, for various spin periods and arbitrary inclination. To this end, the time-dependent Maxwell equations are solved in spherical geometry in the force-free approximation using a vector spherical harmonic expansion of the electromagnetic field. An exact analytical enforcement of the divergenceless of the magnetic part is obtained by a projection method. Special care has been given to design an algorithm able to look deeply into the magnetosphere with physically realistic ratios of stellar $R_*$ to light-cylinder $\rlight$ radius. We checked our code against several analytical solutions, like the Deutsch vacuum rotator solution and the Michel monopole field. We also retrieve energy losses comparable to the magneto-dipole radiation formula and consistent with previous similar works. Finally, for arbitrary obliquity, we give an expression for the total electric charge of the system. It does not vanish except for the perpendicular rotator. This is due to the often ignored point charge located at the centre of the neutron star. It is questionable if such solutions with huge electric charges could exist in reality except for configurations close to an orthogonal rotator. The charge spread over the stellar crust is not a tunable parameter as is often hypothesized.

Exact analytical expansion of an off-axis Gaussian laser beam using the translation theorems for the vector spherical harmonics

The interaction of a Gaussian laser beam with a particle that is located off axis is a fundamental problem encountered across many scientific fields, including biological physics, chemistry, and medicine. For spherical geometries, generalized Lorenz-Mie theory affords a solution of Maxwell's equations for the scattering from such a particle. The solution can be obtained by expanding the laser fields in terms of vector spherical harmonics (VSHs). However, the computation of the VSH expansion coefficients for off-axis beams has proven challenging. In the present study, we provide a very viable, theoretical framework to efficiently compute the sought-after expansion coefficients with high numerical accuracy. We use the existing theory for the expansion of an on-axis laser beam and employ Cruzan's translation theorems [Q. Appl. Math.20, 33 (1962)] for the VSHs to obtain a description for more general off-axis beams. The expansion coefficients for the off-axis laser beam are presented in an analytical form in terms of an infinite series over the underlying translation coefficients. A direct comparison of the electromagnetic fields of such a beam expansion with the original laser fields and with results obtained using numerical quadratures shows excellent agreement (relative errors are on the order of ≲10(-3). In practice, the analytical approach presented in this study has numerous applications, reaching from multiparticle scattering problems in atmospheric physics and climatology to optical trapping, sorting, and sizing techniques.

Wavelength-dependent intensity distribution of a Gaussian beam scattered by a spherical particle

The wavelength-dependent intensity of a light beam scattered by an isotropic homogeneous particle illuminated with an on-axis monochromatic polarized Gaussian beam is calculated in this paper. The vector spherical harmonics expansion and T transformation matrix are used to form the theoretical basis. Numerical results show that the angular intensity distribution is symmetric for an on-axis polarized Gaussian beam illumination upon an aqueous spherical particle at 473, 532 and 660 nm. For the specified aqueous spherical particle, the scattered intensity distribution decreases with increase in wavelengths.

Scalar and vector spherical harmonic spectral equations of rotating magnetohydrodynamics

SUMMARY Vector spherical harmonic analyses have been used effectively to solve laminar and mean-field magnetohydrodynamic dynamo problems with product interactions, such as magnetic induction, anisotropic alpha-effect and anisotropic magnetic diffusion, that are difficult to analyse spectrally in spherical geometries. Spectral forms of the non-linear rotating, Boussinesq and anelastic, momentum, magnetic induction and heat equations are derived for spherical geometries from vector spherical harmonic expansions of the velocity, magnetic induction, vorticity, electrical current and gravitational acceleration and from scalar spherical harmonic expansions of the pressure and temperature. By combining the vector spherical harmonic spectral forms of the momentum equation and the magnetic induction equation with poloidal–toroidal representations of the velocity and the magnetic field, non-linear spherical harmonic spectral equations are also derived for the poloidal–toroidal potentials of the velocity or the momentum density in the anelastic approximation and the magnetic field. Both compact and spectral interaction expansion forms are given. Vector spherical harmonic spectral forms of the linearized rotating magnetic induction, momentum and heat equations for a general basic state can be obtained by linearizing the corresponding non-linear spectral equations. Similarly, the spherical harmonic spectral equations for the poloidal–toroidal potentials of the velocity and the magnetic field may be linearized. However, for computational applications, new alternative hybrid linearized spectral equations are derived. The algorithmically simpler hybrid equations depend on vector spherical harmonic expansions of the velocity, magnetic field, vorticity, electrical current and gravitational acceleration of the basic state and scalar spherical harmonic expansions of the poloidal–toroidal potentials of the perturbation velocity, magnetic field and temperature. The spectral equations derived herein may be combined with the corresponding spectral forms of anisotropic diffusion terms.

Exact solution for the scattering and absorption properties of sphere clusters on a plane surface

A formulation and computational scheme are presented for predicting the scattering and absorption cross-sections, and the scattering matrix elements, of clusters of non-intersecting spheres that are lying on or above an infinite plane surface and exposed to plane-wave radiation. The formulation provides an exact solution to Maxwell's equations and the associated boundary conditions on the spheres and the plane surface, and is applicable for arbitrary refractive indices for the spheres and the surface. A simplified strategy is presented for the calculation of the surface reflection matrix, which transforms the reflected scattered field from one sphere into a regular vector spherical harmonic expansion centered about another sphere. The calculation results are presented for the clusters of one, two, and four polystyrene spheres, with size parameters of one and 10, lying on a silicon substrate, and are compared with the predictions from the normal incidence approximation (NIA) in which the reflectance of the surface is assumed constant at the normal incidence value. The results show that the accuracy of the NIA is highly dependent on the extent of the sphere cluster, the angle of incidence, and the particular quantity (cross-sections, scattering matrix elements) under examination.

Mie scattering by an anisotropic object Part I Homogeneous sphere

Establishing a vector spherical harmonic expansion of the electromagnetic field propagating inside an arbitrary anisotropic medium, we extend Mie theory to the diffraction by an anisotropic sphere, with or without losses. The particular case of a uniaxial material leads to a simpler analysis. This work opens the way to the construction of a differential theory of diffraction by a three-dimensional object with arbitrary shape, filled by an arbitrary anisotropic material.

Invertibility of current density from near-field electromagnetic data

The problem of determining a current density confined to a volume from measurements of the magnetic and electric fields it produces exterior to that volume is known to have nonunique solutions. Despite the nonuniqueness of the inversion we show that one may nevertheless uniquely determine certain moments of the vector spherical harmonic expansion of the current. It is demonstrated that the determination of these moments allows for the unique inversion of a current density confined to a spherical shell. Although unique the inversion may be ill conditioned and require a regularization of the inversion as demonstrated in an example numerical inversion.

A vector spherical harmonic spectral code for linearised magnetohydrodynamics

Linearised rotating magnetohydrodynamic stability code for the steady axisymmetric basic states of an electrically conducting fluid sphere is described. The code generates compact hybrid angular spectral forms of the magnetic induction, heat and Boussinesq Navier-Stokes equations, using toroidal and poloidal representations of the perturbation vector fields, and vector or scalar spherical harmonic expansions of all fields. The momentum equation may include inertial, Coriolis, buoyancy, viscous and magnetic Lorentz forces. Three subroutines evaluate the spectral interactions of products. There are only six radial functions, which are discretised using uniform second-order finite differences. The resulting large scale complex non-hermitian generalised eigen- and critical-value problems are solved using inverse and Newton-Raphson iteration methods, respectively. Test results are presented for several models.

Calculation of total cross sections of multiple-sphere clusters

A method for calculating the extinction, absorption, and scattering cross sections of clusters of neighboring spheres for both fixed and random orientations is developed. The analysis employs the superposition formulation for radiative interactions among spheres, in which the total field from the cluster is expressed as a superposition of vector spherical harmonic expansions about each of the spheres in the cluster. Through the use of addition theorems a matrix equation for the expansion coefficients is obtained. Further application of addition theorems on the inverse of the coefficient matrix is shown to yield analytical expressions for the orientation-averaged total cross sections of the sphere cluster. Calculations of the cross sections of pairs of spheres and fractal aggregates of several spheres are presented. It is found that a dipole representation of the field in each sphere does not adequately predict the absorption cross section of clusters of small-size-parameter spheres when the spheres are highly conducting. For this situation several multipole orders are required for an accurate calculation of the absorption cross section. In addition, the predicted absorption of sphere clusters can be significantly greater than that estimated from the sum of the isolated-sphere cross sections.