Spherical Harmonics Research Articles

Electromagnetic scattering by curved surfaces and calculation of radiation force: Lattice Boltzmann simulations

This study aims to investigate the effectiveness of the lattice Boltzmann method (LBM) in studying the scattering of electromagnetic waves by curved and complex surfaces. The computation of Maxwell’s equations is done by solving for a pair of distribution functions, which evolve based on a two-step process of collision and streaming. LBM bypasses the need for expansion via vector spherical harmonics and thus is amenable to scatterers with complex geometries. We have employed LBM to compute the scattering width and radiation force for perfect electrically conducting and dielectric cylinders of circular and elliptical cross sections. Both smooth and corrugated surfaces are studied, and the results are compared against known analytical and numerical solutions from other methods. To ensure the broad applicability of the method, we have explored a wide range of parameter space—the dielectric constant and particle size to the wavelength ratio spanning Rayleigh, Mie, and geometrical optics regimes. Our simulations have successfully reproduced well-known analytical and numerical solutions, confirming the accuracy and reliability of the LBM for scattering calculations by complex-shaped objects.

Vector random fields on the arccos-quasi-quadratic metric space

An arccos-quasi-quadratic metric is defined as the composition of arccosine and quasi-quadratic functions over a subset of R d + 1 such as a ball, an ellipsoid, a simplex or a corner of the d-cube, a bounded solid cone, a bounded solid hyperboloid, or their surfaces. This metric is not only conditionally negative definite (or of negative type) but also a measure definite kernel, and the metric space incorporates several important cases in a unified framework so that we are able to study metric-dependent random fields on different metric spaces in a unified manner. This paper constructs the Gaussian vector random field on the arccos-quasi-quadratic metric space via an infinite series expression based on spherical harmonics, introduces a class of elliptically contoured vector random fields with metric-dependent covariance matrix functions that possess the ultraspherical polynomial expressions, and studies their sample path properties. In particular, the property of strong local nondeterminism and Hölder continuity are established, and Abelian and Tauberian theorems are derived.

Machine learning approaches for modelling of molecular polarizability in gold nanoclusters

The polarizability of molecules describes their response to an external electric field. It quantifies the ability of a system to form an induced dipole moment when subjected to an electric field. In this work, we investigated isotropic polarizability and anisotropy in the polarizability of gold nanoclusters using various machine-learning algorithms. We utilized high-order invariant descriptors based on spherical harmonics, integrated with machine-learning models like artificial neural network, Gaussian process regression, and kernel ridge regression. Our results demonstrate the efficacy of machine-learning in accurately predicting the polarizability of gold nanoclusters. We find that ANN-based model performs better than the other models.

CunuSHT: GPU Accelerated Spherical Harmonic Transforms on Arbitrary Pixelizations

Abstract We present cunuSHT https://github.com/Sebastian-Belkner/cunuSHT, a general-purpose Python package that wraps a highly efficient CUDA implementation of the nonuniform spin-0 spherical harmonic transform. The method is applicable to arbitrary pixelization schemes, including schemes constructed from equally-spaced iso-latitude rings as well as completely nonuniform ones. The algorithm has an asymptotic scaling of $\mathcal {O}{(\ell _{\rm max}^3)}$ for maximum multipole ℓmax and can be made to achieve machine precision accuracy, considering band-limited transforms for which $N\approx \ell _{\rm max}^2$ (where N is the number of pixels in the map). While cunuSHT is developed for applications in cosmology in mind, it is applicable to various other interpolation problems on the sphere. We outperform the fastest available CPU algorithm at problem sizes ℓmax ∼ 4 · 102 and larger. The speed-up increases with the problem size and reaches a factor of up to 5 for problems with a nonuniform pixelization and ℓmax > 4 · 103 when comparing a single modern GPU to a modern 32-core CPU. This performance is achieved by utilizing the double Fourier sphere method in combination with the nonuniform fast Fourier transform and by avoiding transfers between the host and device. For scenarios without GPU availability, cunuSHT wraps existing CPU libraries. cunuSHT is publicly available and includes tests, documentation, and demonstrations.

Sturm–Liouville problem for a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems

The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier’s separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions.

CGLight: An effective indoor illumination estimation method based on improved convmixer and GauGAN

Illumination consistency is a key factor for seamlessly integrating virtual objects with real scenes in augmented reality (AR) systems. High dynamic range (HDR) panoramic images are widely used to estimate scene lighting accurately. However, generating environment maps requires complex deep network architectures, which cannot operate on devices with limited memory space. To address this issue, this paper proposes CGLight, an effective illumination estimation method that predicts HDR panoramic environment maps from a single limited field-of-view (LFOV) image. We first design a CMAtten encoder to extract features from input images, which learns the spherical harmonic (SH) lighting representation with fewer model parameters. Guided by the lighting parameters, we train a generative adversarial network (GAN) to generate HDR environment maps. In addition, to enrich lighting details and reduce training time, we specifically introduce the color consistency loss and independent discriminator, considering the impact of color properties on the lighting estimation task while improving computational efficiency. Furthermore, the effectiveness of CGLight is verified by relighting virtual objects using the predicted environment maps, and the root mean square error and angular error are 0.0494 and 4.0607 in the gray diffuse sphere, respectively. Extensive experiments and analyses demonstrate that CGLight achieves a balance between indoor illumination estimation accuracy and resource efficiency, attaining higher accuracy with nearly 4 times fewer model parameters than the ViT-B16 model.

Analysis of the three possible event horizons of anti-de Sitter space–time in the Klein–Gordon equation

In this paper, we calculate exactly the solution of Klein–Gordon equation in anti-de Sitter space–time. Due to spherical symmetry, the angular wave functions are given by spherical harmonics. The radial wave functions are given by local Heun functions. We analyze the behavior of the radial wave function in regions near three possible event horizon positions [Formula: see text] and investigate the decay rate, Hawking radiation and Hawking temperature for these three cases. Comparing the results, we verify that if the decay rate and Hawking radiation are greater, then the closer the event horizon is to the singularity of the black hole. Finally, we analyze the special case of a small black hole limit.

Variations of Heat Flux and Elastic Thickness of Mercury From 3‐D Thermal Evolution Modeling

AbstractMercury's low obliquity and 3:2 spin‐orbit resonance create a surface temperature distribution with large latitudinal and longitudinal variations. These propagate via thermal conduction through the thin silicate shell influencing the interior temperature distribution. We use 3‐D thermal evolution models to investigate the effects of lateral variations of surface temperature and crustal thickness on the surface and core‐mantle boundary (CMB) heat fluxes, and elastic thickness distribution. Surface temperature variations cause a long‐wavelength perturbation of the heat fluxes, as well as of the elastic lithosphere thickness, while variations in crustal thickness affect these quantities at small spatial scales. Like the surface temperature, the present‐day CMB heat flux pattern is characterized primarily by spherical harmonics of degrees two and four, which could affect dynamo generation. Placing robust constraints on the thermal evolution based on the available estimates of the age and thickness of the elastic lithosphere remains challenging due to their large uncertainties.

A 3D DEM modeling of soil-rock mixture considering spatial distribution orientation of blocks

Spatial distribution orientations of blocks can cause significant errors in the discrete element model (DEM) calculation of soil-rock mixture (SRM). To avoid this error, spherical harmonic (SH) series whose harmonization degrees fixed at 15 were proposed for block reconstruction. This research refers to the case-history of a deep excavation rift valley spanning from the Mabian to Zhaojue section of the Leshan-Xichang Expressway, mainly containing moderately-weathered silty mudstone, in the Leshan City, Sichuan Province, China. The appropriate degree of finite-term SH series is selected by the volume, surface area. 100 blocks were scanned on site, and sphericity and angularity of the blocks were calculated. The sphericity and angularity of 50 reconstructed blocks were considered for the error analysis of SH method. Moreover, stochastic polyhedron method was considered for comparing different block reconstructions. The maximum block placement angle was defined to control the spatial distribution orientations of the blocks. Large scale direct tests were carried. Numerical simulations of large-scale direct shear tests were conducted to study the influence of the spatial distribution orientations of the blocks on the mechanical properties of the SRMs. The results revealed that the finite-term SH series fixed at 15 accurately reflected the shape characteristics and mechanical behaviors of actual blocks. The spatial distribution orientations of the blocks had a minimal impact on the friction angle and cohesion of SRM constructed through the SH method. The SRMs developed via the SH method exhibited marginal variations in contact force and anisotropy index of contact across diverse block placement strategies. The evolution of coordination number was closer when employing the SH method under varied block placement methods. Blocks reconstructed by the SH method, could mitigate errors in DEM calculation caused by the spatial distribution orientations of the blocks.

Embedding space approach to Lorentzian CFT amplitudes and causal spherical functions

Conformal field theory in a Minkowski setting is discussed in an embedding space approach, paying special attention to causality constraints for four-point amplitudes. The physics of dilatation and Lorentz boost is emphasized in specifying the noncompact maximal Abelian subgroup of SO(d,2). Reduction of a conformal field theory four-point amplitudes as functions of cross ratios is shown to be equivalent to enforcing H bi-invariance, i.e., F(hgh′)=F(g), with g∈SO(d,2) and H an appropriate subgroup. Causality is imposed by introducing appropriate semigroups. Causal zonal spherical functions are constructed, making contact with Minkowski conformal blocks introduced previously. Published by the American Physical Society 2024

FSH3D: 3D Representation via Fibonacci Spherical Harmonics

AbstractSpherical harmonics are a favorable technique for 3D representation, employing a frequency‐based approach through the spherical harmonic transform (SHT). Typically, SHT is performed using equiangular sampling grids. However, these grids are non‐uniform on spherical surfaces and exhibit local anisotropy, a common limitation in existing spherical harmonic decomposition methods. This paper proposes a 3D representation method using Fibonacci Spherical Harmonics (FSH3D). We introduce a spherical Fibonacci grid (SFG), which is more uniform than equiangular grids for SHT in the frequency domain. Our method employs analytical weights for SHT on SFG, effectively assigning sampling errors to spherical harmonic degrees higher than the recovered band‐limited function. This provides a novel solution for spherical harmonic transformation on non‐equiangular grids. The key advantages of our FSH3D method include: 1) With the same number of sampling points, SFG captures more features without bias compared to equiangular grids; 2) The root mean square error of 32‐degree spherical harmonic coefficients is reduced by approximately 34.6% for SFG compared to equiangular grids; and 3) FSH3D offers more stable frequency domain representations, especially for rotating functions. FSH3D enhances the stability of frequency domain representations under rotational transformations. Its application in 3D shape reconstruction and 3D shape classification results in more accurate and robust representations. Our code is publicly available at https://github.com/Miraclelzk/Fibonacci-Spherical-Harmonics.

An optimal ansatz space for moving least squares approximation on spheres

We revisit the moving least squares (MLS) approximation scheme on the sphere Sd-1⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb S^{d-1} \\subset {\\mathbb R}^d$$\\end{document}, where d>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d>1$$\\end{document}. It is well known that using the spherical harmonics up to degree L∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L \\in {\\mathbb N}$$\\end{document} as ansatz space yields for functions in CL+1(Sd-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}^{L+1}(\\mathbb S^{d-1})$$\\end{document} the approximation order OhL+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( h^{L+1} \\right) $$\\end{document}, where h denotes the fill distance of the sampling nodes. In this paper, we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degrees up to L, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as h→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h \\rightarrow 0$$\\end{document}. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space of the sphere as ansatz space.

Wigner state and process tomography on near-term quantum devices

We present an experimental scanning-based tomography approach for near-term quantum devices. The underlying method has previously been introduced in an ensemble-based NMR setting. Here we provide a tutorial-style explanation along with suitable software tools to guide experimentalists in its adaptation to near-term pure-state quantum devices. The approach is based on a Wigner-type representation of quantum states and operators. These representations provide a rich visualization of quantum operators using shapes assembled from a linear combination of spherical harmonics. These shapes (called droplets in the following) can be experimentally tomographed by measuring the expectation values of rotated axial tensor operators. We present an experimental framework for implementing the scanning-based tomography technique for circuit-based quantum computers and showcase results from IBM quantum experience. We also present a method for estimating the density and process matrices from experimentally tomographed Wigner functions (droplets). This tomography approach can be directly implemented using the Python-based software package DROPStomo.

Method for prediction magnetic silencing of uncertain energy-saturated extended technical objects in prolate spheroidal coordinate system

Aim. Development of method for prediction by energy-saturated extended technical objects magnetic silencing based on magnetostatics geometric inverse problems solution and magnetic field spatial spheroidal harmonics calculated in prolate spheroidal coordinate system taking into account of technical objects magnetic characteristics uncertainties. Methodology. Spatial prolate spheroidal harmonics of extended technical objects magnetic field model calculated as magnetostatics geometric inverse problems solution in the form of nonlinear minimax optimization problem based on near field measurements for prediction far extended technical objects magnetic field magnitude. Nonlinear objective function calculated as the weighted sum of squared residuals between the measured and predicted magnetic field COMSOL Multiphysics software package used. Nonlinear minimax optimization problems solutions calculated based on particle swarm nonlinear optimization algorithms. Results. Results of prediction extended technical objects far magnetic field magnitude based on extended technical objects spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system using near field measurements with consideration of extended technical objects magnetic characteristics uncertainty. Originality. The method for prediction by extended technical objects magnetic cleanliness based on spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of magnetic characteristics uncertainty is developed. Practical value. The important practical problem of prediction extended technical objects magnetic silencing based on the spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of extended technical objects magnetic characteristics uncertainty solved. References 48, figures 2.

Evaluation of GOCE/GRACE and combined global geopotential models using GNSS/levelling data over Nigeria

AbstractGlobal Geopotential Models (GGMs) provide valuable information about Earth’s gravity field functionals, such as geoid heights and gravity anomalies. However, ground-based datasets are required to validate these GGMs at the regional and local scales. In this study, we validated the accuracy of GGMs by comparing them with ground-based Global Navigation Satellite System (GNSS)/levelling data for the first time in Nigeria. We employed two validation scenarios: with and without considering spectral consistency using the spectral enhancement method (SEM) to incorporate high and very high frequencies of the gravity field spectrum from the combined global gravity field model (XGM2019e_2159) and the residual terrain model (RTM) derived from the Shuttle Radar Topography Mission (SRTM) data, respectively. The results of this evaluation confirmed that the application of SEM improved the assessment of the GGM solutions in an unbiased manner. Integrating XGM2019e_2159 and SRTM data to constrain the high-frequency component of geoid heights in Gravity Field and Steady-State Ocean Circulation Explorer (GOCE)-based GGMs led to an improvement of approximately 10% in reducing the standard deviation (STD) relative to when SEM was not applied. GO_CONS_GCF_2_TIM_R6 at spherical harmonics (SH) of up to degree and order 260 demonstrated the lowest STD when compared to GO_CONS_GCF_2_DIR_R6 and GO_CONS_GCF_2_SPW_R5, with a reduction from 0.380 m without SEM application to 0.342 m with SEM implementation. In addition, four transformation models, namely, linear, four-parameter, five-parameter, and seven-parameter models, were evaluated. The objective is to mitigate the reference system offsets between the GNSS/levelling data and the GGMs and to identify the particular parametric model with the smallest STD across all GGMs. This effort reduced the GGMs misfits to GNSS/levelling to 0.30 m, representing a 15.3% decrease in STD. Notably, the XGM2019e_2159 model provides this improvement.

Comparison of terrestrial exospheric hydrogen 3D distributions at solar minimum and maximum using TWINS Lyman-α observations

Remote sensing observations of far-ultraviolet (FUV) emissions have been used to estimate 3D neutral hydrogen (H) density models of the terrestrial exosphere under solar minimum (2008) and solar maximum (2013 and 2015) conditions. Specifically, we used Lyman-alpha (Lyman-α, FUV at 121.6 nm) radiance data acquired by the Lyman-alpha detectors (LADs) onboard NASA’s TWINS satellites, collected over several days for each of the 3 years to provide sufficient coverage of the exospheric region. The datasets included Lyman-α measurements taken only above Earth radii (Re) of 3.75, assumed to be the optically thin region of the exosphere, where the measured Lyman-α intensity along a line of sight (LOS) is linearly proportional to the atomic hydrogen column density. Based on the calibration using multiple UV-bright stars, a significant decrease in TWINS1 LAD1/2 sensitivities was found for 2013 (LAD2 ∼1/2, LAD ∼1/3) and 2015 (LAD2 ∼1/3, LAD ∼1/7) compared to 2008. The calibration uncertainty was derived to be, on average, 5%. We estimated the 3D global hydrogen density distributions from these radiance data using two different tomographic inversion methods: first, a parametric fitting method based on a spherical harmonic function of order 3 and second, a high degree-of-freedom retrieval approach to validate our retrievals of H density. Both inversion methods consistently incorporate the effects of Lyman-α absorption within the exosphere and Lyman-α re-emission from Earth’s albedo. Our results reveal that H densities during solar maximum conditions are, on average, 30%–40% higher than those during solar minimum. All models showed a high concentration of atomic H on the dayside and nightside near the Sun–Earth line, which determines a nose/geotail structure consistent with theoretical effects from the solar radiation pressure. Furthermore, we identified that H-density enhancements during solar maximum with respect to solar minimum conditions occur at mid-to-high latitudes, particularly on the dusk side, while no significant enhancement seems to occur near the dayside nose.

Low-degree gravity field coefficients based on inverse GNSS method: insights into hydrological and ice mass change studies

The relative displacements of stations from a global network of Global Navigation Satellite System (GNSS) sites provide information on global mass transport. In this study, we use 19 years of global GNSS station displacements from the 3rd International GNSS Service reprocessing campaign to estimate the coefficients of the spherical harmonics of the Earth’s gravity field up to degree and order 8 using the inverse GNSS method based on elastic loading theory. The results indicate that the C30 coefficient can be derived based on GNSS station displacements as an alternative to solutions provided by Satellite Laser Ranging (SLR) and Gravity Recovery and Climate Experiment (GRACE). GNSS may support GRACE solutions that face the problems of deriving C30, which has fundamental meaning in estimating ice mass changes in polar regions. The recovery of Antarctic ice sheet mass change from January 2007 to December 2020 based on coefficients replaced by GNSS estimates results in a linear trend of − 152 ± 4 Gt/year, in comparison to − 149 ± 2 Gt/year for the replacement based on SLR from GRACE Technical Note #14. The results indicate that the spatial and seasonal patterns of terrestrial water storage changes derived from GNSS are consistent with those estimated using GRACE/GRACE Follow-On and SLR at a few-millimeter level in the Amazon and Brahmaputra River basin regions.

Calculation of the Gaunt Coefficients over Real Spherical Harmonics

Calculation of the Gaunt Coefficients over Real Spherical Harmonics

Analysis on scattering and inner near-field characteristics of a uniaxial anisotropic sphere by an off-axis high-order Bessel (vortex) beam

Based on the generalized Lorenz–Mie theory (GLMT) and the Fourier transform method, a theoretical approach is introduced to study the scattering of a uniaxial anisotropic sphere illuminated by an off-axis high-order Bessel (vortex) beam (HOBVB). According to the orthogonality of the associated Legendre function and exponential function, a concise expression of the expansion coefficients of the off-axis HOBVB in terms of the spherical vector wave functions (SVWFs) is derived that can effectively reconstruct the HOBVB with all conical angles. The differences of scattering characteristics of a uniaxial anisotropic sphere illuminated by an on-axis and off-axis HOBVB and a plane wave are exhibited. Influences of the topological charge, conical angle, particle size, and off-axis distance on the angle distributions of the radar cross-section (RCS), scattering and extinction efficiencies, and asymmetric factor are analyzed in detail. The unique internal and near-field distributions of a uniaxial anisotropic spherical particle illuminated by an on-axis and off-axis HOBVB are demonstrated. The results provide insights into the scattering and Bessel beam–matter interactions and may find important applications in optical propagation and optical micromanipulation, microwave engineering, target shielding, and near-field measurement.

Methodology for 3D sound synthesis of directional acoustic sources by higher-order ambisonics

This paper presents the 3D sound field synthesis using both the multimodal method and higher-order ambisonics (HOA) of the pressure field radiated by directional acoustic sources. Ambisonics is a technique for encoding and reproducing measured or modeled (virtual) sound pressure field, based on a decomposition of the acoustic field over spherical harmonics. The directional source considered in this work is an acoustic horn excited by a flat piston. The free-field radiation from this horn is first modeled accurately over a wide frequency range using the multimodal method, which requires relatively low computational resources. This radiated pressure field, collected on a dual-layer sphere of virtual sensors distributed over a Lebedev geometry, allows its projection into the ambisonic domain. The pressure field is then synthetized in the laboratory's 3D 5th order HOA spatialization sphere, which consists of fifty-six loudspeakers. This offers the ability of listening to the radiated sound using a higher-order ambisonic synthesis of a 'virtual' source before it is manufactured. To qualitatively evaluate the performance of the proposed procedure, the transfer function of the synthesized horn is measured around the listening point within the spatialization sphere. Key words : 3D sound synthesis, high-order ambisonics, directive sources, waveguides, horn, multimodal method