Terms Of Spherical Harmonics Research Articles

Efficient multipole representation for matter-wave optics

Technical optics with matter waves requires a universal description of three-dimensional traps, lenses, and complex matter-wave fields. In analogy to the two-dimensional Zernike expansion in beam optics, we present a three-dimensional multipole expansion for Bose-condensed matter waves and optical devices. We characterize real magnetic chip traps, optical dipole traps, and the complex matter-wave field in terms of spherical harmonics and radial Stringari polynomials. We illustrate this procedure for typical harmonic model potentials as well as real magnetic and optical dipole traps. Eventually, we use the multipole expansion to characterize the aberrations of a ballistically interacting expanding Bose–Einstein condensate in (3 + 1) dimensions. In particular, we find deviations from the quadratic phase ansatz in the popular scaling approximation. The scheme is data efficient by representing millions of complex amplitudes of a field on a Cartesian grid in terms of a low order multipole expansion without precision loss. This universal multipole description of aberrations can be used to optimize matter-wave optics setups, for example, in matter-wave interferometers.

A mechanical model of ocular bulb vibrations and implications for acoustic tonometry.

In this study, we propose a comprehensive mechanical model of ocular bulb vibrations and discuss its implications for acoustic tonometry. The model describes the eye wall as a spherical, pre-stressed elastic shell containing a viscoelastic material and accounts for the interaction between the elastic corneoscleral shell and the viscoelastic vitreous humor. We investigate the natural frequencies of the system and the corresponding vibration modes, expanding the solution in terms of scalar and vector spherical harmonics. From a quantitative point of view, our findings reveal that the eyebulb vibration frequencies significantly depend on IOP. This dependency has two origins: "geometric" stiffening, due to an increase of the pre-stress, and "material" stiffening, due to the nonlinearity of the stress-strain curve of the sclera. The model shows that the second effect is by far dominant. We also find that the oscillation frequencies depend on ocular rigidity, but this dependency is important only at relatively large values of IOP. Thus close to physiological conditions, IOP is the main determinant of ocular vibration frequencies. The vitreous rheological properties are found to mostly influence vibration damping. This study contributes to the understanding of the mechanical behavior of the eye under dynamic conditions and thus has implications for non-contact intraocular pressure measurement techniques, such as acoustic tonometry. The model can also be relevant for other ocular pathological conditions, such as traumatic retinal detachment, which are believed to be influenced by the dynamic behavior of the eye.

Scattering of elastic waves by a sphere with orthorhombic anisotropy and application to polycrystalline material characterization

Scattering of elastic waves by an anisotropic sphere with orthorhombic symmetry inside an isotropic medium is studied and applied to characterization of polycrystalline materials with anisotropic grains. For a single sphere the waves in the isotropic surrounding are expanded in the spherical vector wave functions. Inside the sphere, the elastodynamic equations are first transformed to spherical coordinates and the displacement field is expanded in terms of the vector spherical harmonics in the angular directions and power series in the radial direction. The governing equations inside the sphere give recursion relations among the expansion coefficients in the power series. The boundary conditions on the sphere then determine the relation among the scattered wave expansion coefficients and those of the incident wave, expressed as the transition (T) matrix. For low frequencies the elements of the T matrix are obtained in explicit form. According to the theory of Foldy the T matrix elements of a single sphere are used to study attenuation and phase velocity of polycrystalline materials, explicitly for low frequencies. Comparisons of the present method with previously published results and recent FEM results show a good correspondence for low frequencies. The present approach shows a better agreement with FEM for strongly anisotropic materials in comparison with other published methods.

Sound radiation of a vibrating circular plate set in a hemispherical enclosure

This study rigorously addresses the Neumann–Robin problem of sound radiation from a vibrating flexible disk embedded into a hemispherical enclosure. The enclosure is acoustically soft on its inside surface and hard on its outside surface. The method of superimposed solutions, expressed in terms of Dini series and spherical harmonics, has been applied for this purpose. The presented solution is applicable to any vibration velocity profile on the radiator, with an elastically supported thin plate excited asymmetrically used as an example. The results presented here can be valuable for modeling vibroacoustic responses of various speakers, sensors, and microphones with edges either clamped to the enclosure or nearly free. Achieving this involves modifying the values of the boundary stiffness for resisting transverse displacement and the rotation of the plate’s edge. Additionally, when applying these results to model the responses of alarm or signaling transducers, high-amplitude responses are expected for selected resonant frequencies. In such cases, the boundary stiffness values can be set to ensure multiple strong and wide resonant maxima in the frequency characteristics. In addition, the results presented in this study include solving the dynamic equation of motion for the plate or rigid piston with external excitation. This equation is coupled with the wave equation inside and outside the hemispherical enclosure, allowing the inclusion of the enclosure’s effect on the plate-enclosure system’s responses. This approach can be useful for modeling and improving the acoustic responses of spherical speakers. In such cases, the boundary stiffness values should be relatively small to emulate a soft suspension of the plate. The impedance of the enclosure’s internal surface can be arbitrarily selected, ranging from acoustically hard to acoustically soft. In this study, the impedance of mineral wool has been applied. Numerical analysis is presented for frequencies below 4kHz, with errors smaller than 0.2%. The results have been validated using the finite element method.

Modal Projection for Quasi-Homogeneous Anisotropic Turbulence

This article, or essay, addresses the anisotropic structure and the dynamics of quasi-homogeneous, incompressible turbulence. Modal projection and expansions in terms of spherical harmonics in three-dimensional Fourier space are in line with a seminal study by Jack Herring, around the so-called Craya–Herring frame of reference, with a large review of the related approaches to date. The research part is focused on structure and dynamics of rotating sheared turbulence, including a description of both directional and polarization anisotropy with a minimal number of modes. Effort is made to generalize expansions in terms of scalar spherical harmonics (SSHs) to vector spherical harmonics (VSHs). Looking at stochastic fields, for possibly intermittent vector fields, some directions are explored to reconcile modal projection, firstly used for smooth vector fields, and multifractal approaches for internal intermittency but far beyond scalar correlations, such as structure functions. In order to illustrate turbulence from Earth to planets, stars, and galaxies, applications to geophysics and astrophysics are touched upon, with generalization to coupled vector fields (for kinetic, magnetic, and potential energies), possibly dominated by waves (Coriolis, gravity, and Alfvén).

Dirac equation in relativistic density functional theory and mapped Fourier grid method

For studying the electronic structure of solids consisting of heavy atoms, the Dirac–Kohn–Sham equation is considered in the presence of the Mathieu potential. The two-component spinors and the potential function are expanded in terms of spherical harmonics. Also, a numerical approach is presented to study the Dirac–Kohn–Sham equation in the presence of the noncentral Mathieu potential. Then, the energy eigenvalues and eigenvectors are obtained in the case of spherically expanded potential in the Brillouin zoon by using the mapped Fourier grid method.

Orientation-angle-resolved photoelectron angular distribution in dissociative ionization of methanol induced by an intense ultraviolet laser pulse

We investigate dissociative ionization of methanol, $\mathrm{C}{\mathrm{H}}_{3}\mathrm{OH}\ensuremath{\rightarrow}{\mathrm{CH}}_{3}{}^{+}+\mathrm{OH}+{e}^{\ensuremath{-}}$, in a linearly polarized intense ultraviolet laser field (400 nm, 67 fs, $3.1\ifmmode\times\else\texttimes\fi{}{10}^{13}\phantom{\rule{0.28em}{0ex}}\mathrm{W}/\mathrm{c}{\mathrm{m}}^{2}$) by photoelectron-photoion coincidence three-dimensional momentum imaging. The photoelectron angular distributions (PADs) are recorded as a function of the orientation angle between the molecular C-O bond axis and the laser polarization direction. Photoelectrons are emitted dominantly along the laser polarization direction when ${\mathrm{CH}}_{3}{\mathrm{OH}}^{+}$ is produced in the $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{X}$ state via the four-photon absorption. By contrast, the photoelectron emission becomes prominent along the direction perpendicular to the laser polarization direction when ${\mathrm{CH}}_{3}{\mathrm{OH}}^{+}$ is produced in the $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}$ state via the five-photon absorption. The orientation-angle-resolved PADs are expanded in terms of spherical harmonics in the laboratory frame, and then the anisotropy of the PADs varying as a function of the orientation angle of the C-O bond axis with respect to the laser polarization direction is examined quantitatively. We discuss how the shape of the molecular orbital from which an electron is emitted is reflected in the recorded PADs.

Planet Eclipse Mapping with Long-term Baseline Drifts

High-precision lightcurves combined with eclipse-mapping techniques can reveal the horizontal and vertical structure of a planet’s thermal emission and the dynamics of hot Jupiters. Someday, they even may reveal the surface maps of rocky planets. However, inverting lightcurves into maps requires an understanding of the planet, star, and instrumental trends because they can resemble the gradual flux variations as the planet rotates (i.e., partial phase curves). In this work, we simulate lightcurves with baseline trends and assess the impact on planet maps. Baseline trends can be erroneously modeled by incorrect astrophysical planet map features, but there are clues to avoid this pitfall in both the residuals of the lightcurve during eclipse and sharp features at the terminator of the planet. Models that use a Gaussian process or polynomial to account for a baseline trend successfully recover the input map even in the presence of systematics but with worse precision for the m = 1 spherical harmonic terms. This is also confirmed with the ThERESA eigencurve method where fewer lightcurve terms can model the planet without correlations between the components. These conclusions help aid the decision on how to schedule observations to improve map precision. If the m = 1 components are critical, such as measuring the east/west hot-spot shift on a hot Jupiter, better characterization of baseline trends can improve the m = 1 terms’ precision. For latitudinal north/south information from m ≠ 1 mapping terms, it is preferable to obtain high signal to noise at ingress/egress with more eclipses.

Scattering of elastic waves by a sphere with cubic anisotropy with application to attenuation in polycrystalline materials

Scattering of elastic waves by an anisotropic sphere with cubic symmetry inside an isotropic medium is studied. The waves in the isotropic surrounding are expanded in the spherical vector wave functions. Inside the sphere, the elastodynamic equations are first transformed to spherical coordinates and the displacement field is expanded in terms of the vector spherical harmonics in the angular directions and a power series in the radial direction. The governing equations inside the sphere give recursion relations among the expansion coefficients in the power series. The boundary conditions on the sphere then determine the expansion coefficients of the scattered wave. This determines the transition ( T ) matrix elements which are calculated explicitly to the leading order for low frequencies. Using the theory of Foldy, the T matrix elements of a single sphere are used to study attenuation and phase velocity of polycrystalline materials with cubic symmetry, explicitly for low frequencies and numerically for intermediate frequencies. Numerical comparisons of the present method with previously published results and recent finite element method (FEM) results show a good correspondence for low and intermediate frequencies. The present approach shows a better agreement with FEM for strongly anisotropic materials in comparison with other published methods.

Time-step heat problem on the mesh: asymptotic behavior and decay rates

Abstract In this article, we study the asymptotic behavior and decay of the solution of the fully discrete heat problem. We show basic properties of its solutions, such as the mass conservation principle and their moments, and we compare them to the known ones for the continuous analogue problems. We present the fundamental solution, which is given in terms of spherical harmonics, and we state pointwise and ℓ p {\ell^{p}} estimates for that. Such considerations allow to prove decay and large-time behavior results for the solutions of the fully discrete heat problem, giving the corresponding rates of convergence on ℓ p {\ell^{p}} spaces.

Generalized form of the magnetic anisotropy field in micromagnetic and atomistic spin models

We present a general approach to the derivation of the effective anisotropy field which determines the dynamical behaviour of magnetic spins according to the Landau-Lifshitz-Gilbert equation. The approach is based on the gradient in spherical polar coordinates with the final results being expressed in Cartesian coordinates as usually applied in atomistic and micromagnetic model calculations. The approach is generally valid for all orders of anisotropies including higher order combinations of azimuthal and rotational anisotropies often found in functional magnetic materials such as permanent magnets and an emerging class of antiferromagnetic materials with applications in spintronics. Anisotropies are represented in terms of spherical harmonics which have the important property of rational temperature scaling. Effective field vectors are given for anisotropies up to sixth order, presenting a unified framework for implementing higher order magnetic anisotropies in numerical simulations.

Magnetic neutron scattering from spherical nanoparticles with Néel surface anisotropy: analytical treatment.

The magnetization profile and the related magnetic small-angle neutron scattering cross section of a single spherical nanoparticle with Néel surface anisotropy are analytically investigated. A Hamiltonian is employed that comprises the isotropic exchange interaction, an external magnetic field, a uniaxial magnetocrystalline anisotropy in the core of the particle and the Néel anisotropy at the surface. Using a perturbation approach, the determination of the magnetization profile can be reduced to a Helmholtz equation with Neumann boundary condition, whose solution is represented by an infinite series in terms of spherical harmonics and spherical Bessel functions. From the resulting infinite series expansion, the Fourier transform, which is algebraically related to the magnetic small-angle neutron scattering cross section, is analytically calculated. The approximate analytical solution for the spin structure is compared with the numerical solution using the Landau-Lifshitz equation, which accounts for the full nonlinearity of the problem. The signature of the Néel surface anisotropy can be identified in the magnetic neutron scattering observables, but its effect is relatively small, even for large values of the surface anisotropy constant.

Describing the characteristics of light field in architectural spaces using spherical harmonic function

The direction, density, and diffusivity of light are important indicators of spatial characteristics in describing a three-dimensional light environment. Mury presented a method for describing, measuring, and visualizing the structure of light fields using spherical harmonics in terms of changes in the density and direction of light in three-dimensional space. We extended this study by using higher-order spherical harmonics, which would represent more diverse characteristics of the light environment. We also quantitatively described the light environment as numerical values and investigated the correspondence between these numerical values and human perceptual quantities. As a result, we confirmed that there is a certain degree of correspondence between the “complexity” quantified by the spherical harmonic and the “complexity” perceived by people when observing real space.

Nondegeneracy of the positive solutions for critical nonlinear Hartree equation in ℝ⁶

We prove that any positive solution for the critical nonlinear Hartree equation − △ ⁡ u ( x ) − ∫ R 6 | u ( y ) | 2 | x − y | 4 d y u ( x ) = 0 , x ∈ R 6 , \begin{equation*} -\operatorname {\bigtriangleup }{u}\left (x\right ) -\int _{\mathbb {R}^6} \frac {\left |{u}\left (y\right )\right |^2 }{ \left |x-y\right |^4 }\mathrm {d}y \,{u}\left (x\right )=0,\quad \quad x\in \mathbb {R}^6, \end{equation*} is nondegenerate. Firstly, in terms of spherical harmonics, we show that the corresponding linear operator can be decomposed into a series of one dimensional linear operators. Secondly, by making use of the Perron-Frobenius property, we show that the kernel of each one dimensional linear operator is finite. Finally, we show that the kernel of the corresponding linear operator is the direct sum of the kernel of all one dimensional linear operators.

Timing-residual power spectrum of a polarized stochastic gravitational-wave background in pulsar-timing-array observation

We study the observation of a stochastic gravitational-wave background (SGWB) made by a pulsar-timing array in the spherical harmonic space of the observable. The observable is a timing residual which is the time-averaged redshift fluctuation of a pulsar over the duration of observation. Using the Sachs-Wolfe line-of-sight integral for the redshift fluctuation, we derive the power spectrum of the timing residual, from which we develop a fast algorithm to compute the overlap reduction functions for the SGWB intensity and polarization anisotropies. We find that the algorithm is less complicated and more efficient than our previous work which is based on the expansion of the polarization basis tensors in terms of spherical harmonics. Also, we use the power spectrum to construct the bipolar spherical harmonic coefficients that characterize the statistical isotropy of the SGWB. In particular, the coefficients for the linear-polarization anisotropy are worked out for the first time. Our harmonic-space method is useful for the data analysis in future pulsar-timing-array observation on a large number of pulsars as well as for the measurement of the statistical isotropy of the SGWB.

Multilevel representations of isotropic Gaussian random fields on the sphere

Abstract Series expansions of isotropic Gaussian random fields on $\mathbb {S}^2$ with independent Gaussian coefficients and localized basis functions are constructed. Such representations with multilevel localized structure provide an alternative to the standard Karhunen–Loève expansions of isotropic random fields in terms of spherical harmonics. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field. In addition, numerical illustrations are given and an application to random elliptic PDEs on the sphere is analysed.

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.

Pulsar-timing measurement of the circular polarization of the stochastic gravitational-wave background

Pulsar-timing arrays (PTAs) are in the near future expected to detect a stochastic gravitational-wave background (SGWB) produced by a population of inspiralling supermassive black hole binaries. In this work, we consider a background that can be anisotropic and circularly polarized. We use the expansion of the intensity and the circular polarization in terms of spherical harmonics and the overlap reduction functions for each term in this expansion. We propose an unbiased real-space estimator that can separate the intensity and circular-polarization contributions of the SGWB to pulsar-timing-residual correlations and then validate the estimator on simulated data. We compute the signal-to-noise ratio of a circular-polarization component that has a dipole pattern under different assumptions about the PTA. We find that a nearly maximal circular-polarization dipole may be detectable, which can aid in determining whether or not the background is dominated by a handful of bright sources.

A characterisation of the continuum Gaussian free field in arbitrary dimensions

We prove that under certain mild moment and continuity assumptions, the d-dimensional continuum Gaussian free field is the only stochastic process satisfying the usual domain Markov property and a scaling assumption. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.

Active shim coils design for Halbach magnet based on inverse boundary element method

With advantages of small volume, high field strength, and compact stray field, the Halbach array magnet is of great potential in desktop NMR applications. However, field inhomogeneity poses a main obstacle to its applications. Active shimming is necessary for improving the field homogeneity. An active shim coil design method for cylindrical Halbach magnet based on inverse boundary element method (iBEM) is introduced in this work. Shim coils targeting 1st∼3rd and Z4/Z5/Z6 spherical harmonic (SH) terms were designed with constraints of field accuracy, resistance, inductance, and current density. Wire loops and symmetric property influence were discussed to further improve the performance of high-order shim coils. Finally, with the prototype of the shim coils constructed, field pattern and SH transformation proved that the coils designed could fit the target SH distribution, though contamination from low-order components existed in high-order terms. A practical shimming experiment on the Halbach magnet has improved the homogeneity from 40.3 ppm to 3.8 ppm in radius 2.5 mm, height 5 mm cylindrical sample space, verifying the coils' compensation ability. This work provides a flexible inverse coil design method for cylindrical Halbach magnet and discusses the high-order shim coil design and implementation, which is potential in nuclear magnetic resonance applications based on Halbach magnet arrays.