Vector Spherical Harmonics Research Articles

Fundamental Limitations for DOA and Polarization Estimation With Applications in Array Signal Processing

Using a generic electromagnetic model based on the spherical vector harmonics, we provide the Cramer-Rao lower bound (CRB) for direction-on-arrival (DOA) and polarization estimation with arbitrary multiport antennas. By reciprocity, the receiving multiport antenna is characterized in terms of the transmission matrix from incident voltage waves to multipole coefficients of the outgoing spherical vector waves. Explicit results of the CRB for the Gaussian case is given in terms of the transmission matrix, the spherical vector harmonics and its spatial derivatives. Numerical examples are included to illustrate the results

Radiation trapping forces acting on a two-layered spherical particle in a Gaussian beam

Based on the generalized Lorenz-Mie theory (GLMT), the incident fundamental Gaussian beam is expanded in terms of the vector spherical harmonics. We present an expression of radiation trapping forces acting on a multilayered spherical particle in a Gaussian beam by using the theory of electromagnetic momentum, and, as an example, numerical results are given to an axial two-layer absorbing spherical particle, along with a discussion about the influence of the beam waist radius, relative complex refractive index and thickness of the two layers on trapping forces.

Spherical magnetic instabilities in the Earth’s core and equatorial symmetries

The Boussinesq rotating magnetohydrodynamic equations in a conducting fluid sphere, linearised about a steady basic state, are discretised using scalar and vector spherical harmonics. Vector spherical harmonic representations are used for the basic state vector fields as well as the interaction terms in the equations. The solenoidal property of the perturbation magnetic and velocity fields are imposed on the fields and equations using relations between vector spherical harmonics and toroidal–poloidal representations. The method has several distinct advantages over previous discretisation methods: it produces compact angular spectral equations which require only five core subroutines to generate all interactions and product terms, and only six core subroutines to radially discretise the equations using finite-differences; the core subroutines can be extensively tested using only simple models; and its generality permits very flexible code. We apply our approach to the magnetic instabilities of a steady axisymmetric azimuthal magnetic field. To isolate magnetic instabilities and suppress other instabilities the magnetostrophic approximation is made. The perturbation velocity and magnetic field satisfy the inviscid boundary and insulating exterior conditions, respectively. Results are presented for the simplest equatorially symmetric (quadrupole) and equatorially antisymmetric (dipole) magnetic field models which vanish on a single spherical surface. For each model the perturbation fields decouple into two subproblems, in which the perturbation magnetic field has the same or opposite symmetry to the basic state field. A strong preference is found for the equatorially symmetric perturbation magnetic field in the equatorially symmetric model. However, there is only a marginal preference for the equatorial antisymmetric perturbation magnetic field in the equatorially antisymmetric model. We also consider a class of equatorially unsymmetric basic state magnetic fields which are convex combinations of the first two models. Although the critical Elsasser number does not change appreciably, the most unstable mode changes totally from equatorially symmetric to equatorially antisymmetric. Finally, we consider the effect on the equatorially symmetric model of varying the spherical surface on which the field vanishes. The most unstable mode is always equatorially symmetric. When the dimensionless magnetic-zero radius is decreased below about 0.7 the most unstable mode switches from eastward to westward propagation. The magnetic field perturbation in the westward (eastward) case is strongly confined to the region outside (inside) the magnetic-zero radius.

Light diffraction by a three-dimensional object: differential theory

The differential theory of diffraction of light by an arbitrary object described in spherical coordinates is developed. Expanding the fields on the basis of vector spherical harmonics, we reduce the Maxwell equations to an infinite first-order differential set. In view of the truncation required for numerical integration, correct factorization rules are derived to express the components of D in terms of the components of E, a process that extends the fast Fourier factorization to the basis of vector spherical harmonics. Numerical overflows and instabilities are avoided through the use of the S-matrix propagation algorithm for carrying out the numerical integration. The method can analyze any shape and/or material, dielectric or conducting. It is particularly simple when applied to rotationally symmetric objects.

Astrometric Effects of Secular Aberration

One of the main endeavors of fundamental astrometry is to establish a practical realization of an inertial reference frame anchored to celestial objects whose positions are defined in the barycentric coordinates of the solar system. The development of astrometric facilities operating from space at a microarcsecond level of precision makes the non-uniformity of the galactic motion of the barycenter an observable effect that violates the inertiality of the barycentric frame. Most of the observable effect is caused by secular acceleration of the barycenter with respect to the center of the Galaxy. The acceleration results in a pattern of secular aberration which is observable astrometrically as a systematic vector field of the proper motions of distant quasars. We employ the vector spherical harmonics to describe the predicted field of the proper motions and evaluate its amplitude at each point on the celestial sphere. It is shown that the pattern of secular aberration is fully represented by three low-order electric-type vector harmonics, and hence, it is easily distinguishable from the residual rotations of the reference frame and other possible effects, such as the hypothetical long-period gravitational waves. Comprehensive numerical simulations of the grid astrometry with SIM PlanetQuest are conducted. The full covariance matrix of the simulated grid solution is used to evaluate the covariances of the three electric harmonic coefficients, representing the secular aberration pattern of proper motions. We conclude that the grid astrometry with SIM PlanetQuest will be sensitive to the main galactocentric component of secular acceleration, while the peculiar acceleration of the Sun with respect to LSR is expected to be too small to be detected with this astrometric space interferometer.

Angular spectrum representation of the electromagnetic multipole fields, and their reflection at a perfect conductor

Angular spectrum representations are derived for electric and magnetic multipole fields of arbitrary order. The result involves generalized spherical harmonics and generalized vector spherical harmonics, and the representations are in the form of integrals over the k ∥-plane. The representations are especially useful for the study of reflection and transmission of multipole radiation by a plane interface. As an example, we have considered the reflection at a perfect conductor. The reflected field of a multipole field could be expressed in the form of an angular spectrum with a very simple relation to the angular spectrum of the source field. The radiation pattern of a multipole near the perfect conductor is obtained with the method of stationary phase. We also introduce a method for determining the mirror image of the source of an arbitrary multipole.

On the existence of three-dimensional hydrostatic and magnetostatic equilibria of self-gravitating fluid bodies

We develop an analytical spectral method to solve the equations of equilibrium for a self-gravitating, magnetized fluid body, under the only hypotheses that (i) the equation of state is isothermal, (ii) the configuration is scale-free and (iii) the body is electrically neutral. All physical variables are represented as series of scalar and vector spherical harmonics of degree l and order m, and the equilibrium equations are reduced to a set of coupled quadratic algebraic equations for the expansion coefficients of the density and the magnetic vector potential. The method is general, and allows us to recover previously known hydrostatic and magnetostatic solutions possessing axial symmetry. A linear perturbation analysis of the equations in spectral form shows that these basic axisymmetric states, considered as a continuous sequence with the relative amount of magnetic support as control parameter, have in general no neighbouring non-axisymmetric equilibria. This result lends credence to a conjecture originally made by H. Grad, and extends early results obtained by E. Parker to the case of self-gravitating magnetized bodies. The only allowed bifurcations of this sequence of axisymmetric equilibria are represented by distortions with dipole-like angular dependence (l = 1) which can be continued into the non-linear regime. These new configurations are either (i) azimuthally asymmetric (m = ±1) or (ii) azimuthally symmetric but without reflection symmetry with respect to the equatorial plane (m = 0). It is likely that these configurations are not physically acceptable solutions for isolated systems, but represent instead the manifestation of a general gauge freedom of self-similar isothermal systems. To the extent that interstellar clouds can be represented as isolated magnetostatic equilibria, the results of this study suggest that the observed triaxial shapes of molecular cloud cores can be interpreted in terms of weakly damped Alfven oscillations about an equilibrium state.

A Nonlinear Galerkin Scheme Involving Vector and Tensor Spherical Harmonics for Solving the Incompressible Navier--Stokes Equation on the Sphere

This work is concerned with a nonlinear Galerkin method for solving the incompressible Navier--Stokes equation on the sphere. It extends the work of [A. Debussche, T. Dubois, and R. Temam, Theoret. Comput. Fluid Dyn., 7 (1995), pp. 279--315; M. Marion and R. Temam, SIAM J. Numer. Anal., 26 (1989), pp. 1139--1157; J. Shen and R. Temam, Proceedings of the International Conference on Nonlinear Evolution Partial Differential Equations, AMS, Providence, RI, 1997, pp. 363--376] from one-dimensional or toroidal domains to the spherical geometry. In the first part, the method based on type 3 vector spherical harmonics is introduced and convergence is indicated. Further it is shown that the occurring coupling terms involving three vector spherical harmonics can be expressed algebraically in terms of Wigner-$3j$ coefficients. To improve the numerical efficiency and economy we introduce an FFT-based pseudospectral algorithm for computing the Fourier coefficients of the nonlinear advection term. The resulting method scales with $O(N^3)$ if N denotes the maximal spherical harmonic degree. The latter is demonstrated in an extensive numerical example.

Scattering of fundamental Gaussian beam from on-axis cluster spheres

Based on the generalized Lorenz_Mie theory(GLMT),the incident fundamental Gaussi an beam is expanded in terms of the vector spherical harmonics. The interaction of Gaussian beam with a cluster composed of spheres located on the propagation a xis is analyzed. The numerical results are discussed and compared with those obt ained in the plane wave incident case.

Geomagnetism and Schmidt quasi-normalization

SUMMARY Spherical harmonic analysis of the main magnetic field of the Earth and its daily variations is the numerical determination of coefficients of solid spherical harmonics in the mathematical expressions used for the magnetic scalar potential of fields of internal and external origin. The coefficients are determined from vector components of the field and their purpose is to represent the vector field, not to reconstruct the magnetic scalar potential. An alternative interpretation of the spherical harmonic analysis is presented: namely the determination of the coefficients of a series representation of the magnetic vector field on a spherical surface in orthonormal real vector spherical harmonics, which correspond to the internal and external fields, and an additional non-potential toroidal field. The numerical values of the coefficients of an orthonormal vector spherical harmonic series have a direct physical significance, which is not obscured by some arbitrary normalization of the vector spherical harmonics. Therefore, we propose a Schmidt vector normalization to be used in conjunction with the Schmidt quasinormalization of associated Legendre functions. A property of orthonormalized functions is that the standard deviations of the coefficients determined by the method of least squares from ideal data, which are uniformly accurate and uniformly globally distributed, are constant for all coefficients. The real vector spherical harmonic analysis of the geomagnetic field is extended to a spherical shell and conditions that restrict the radial dependence of the vector spherical harmonic coefficients are examined. In particular, two hypotheses for the current systems deriving from the non-potential toroidal component of the magnetic field over the surface of a sphere are presented, namely, Earth‐air currents and field-aligned currents.

Problems involved in elimination of both the singularity in the solution of nonlinear equation of motion on the rotation axis and the unbalance caused by convective heat transfer

We are considering below some complex problems which arise in theoretical description of the processes developing in convective zones of rotating stars. It is common knowledge that the solutions of the starting nonlinear equations of equilibrium or motion may have singularities at the rotation axis, whose elimination in the case of equations of an even more general nature including spins was treated in quantum mechanics by invoking a numerical procedure (see, e.g., Varshalovich et al. (1988)). If spin effects are disregarded, these equations, which describe general representations of conventional and Alfven velocities as expansions in orthogonal vector spherical harmonics can be cast in an analytical form Vandakurov (1999).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Hypothesis on the Origin of Solar Torsional Waves

Latitude distribution of the solar convective zone rotation rate is studied by expanding it in a complete system of orthogonal modes (both symmetric and asymmetric relative to the equator). We find the solution to the problem of determination of the rotation rate of using helioseismic data available for the latitudes of , will also be included into the above latitude list for the reason to be disclosed below. It is known that any rotation rate can always be represented by an expansion in a complete system of orthogonal vector spherical harmonics with a zero superscript (see, e.g., Varshalovich et al. 1988).To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Trigonometric Approximations for the Computation of Radar Cross Sections

Trigonometric approximation methods for the interpolation of the far field of a scatterer lit up by plane waves are presented. The Wacker method, commonly used to determine the far field of an antenna from spherical near field measurements, is adapted to decompose the scattered field in a finite series of vector spherical harmonics. The decomposition coefficients are used to assess the accuracy of the computed field and to efficiently store the far field profile. These coefficients are approximated by trigonometric functions up to a given order which can be estimated from the geometry of the scatterer. In this way an efficient procedure for the interpolation of monostatic and bistatic radar cross sections is obtained.

Toroidal perturbations of Friedmann-Robertson-Walker universes

Explicit expressions are found for the axisymmetric metric perturbations of the closed, flat and open FRW universes caused by toroidal motions of the cosmic fluid. The perturbations are decomposed in vector spherical harmonics on 2-spheres, but the radial dependence is left general. Solutions for general odd-parity $l$-pole perturbations are given for either angular velocities or angular momenta prescribed. In particular, in case of closed universes the solutions require a special treatment of the Legendre equation.

LOW-FREQUENCY SOLUTION FOR A PERFECTLY CONDUCTING SPHERE IN A CONDUCTIVE MEDIUM WITH DIPOLAR EXCITATION

This contribution concerns the interaction of an arbitrarily orientated, time-harmonic, magnetic dipole with a perfectly conducting sphere embedded in a homogeneous conductive medium. A rigorous low-frequency expansion of the electromagnetic field in positive integral powers (jk) to n, k complex wavenumber of the exterior medium, is constructed. The first n = 0 vector coefficient (static or Rayleigh) of the magnetic field is already available, so emphasis is on the calcu- lation of the next two nontrivial vector coefficients (at n = 2 and at n = 3) of the magnetic field. Those are found in closed form from exact solutions of coupled (at n = 2, to the one at n = 0) or uncoupled (at n = 3) vector Laplace equations. They are given in compact fashion, as infinite series expansions of vector spherical harmonics with scalar coefficients (for n = 2). The good accuracy of both in-phase (the real part) and quadrature (the imaginary part) vector components of the diffusive magnetic field are illustrated by numerical computations in a realistic case of mineral exploration of the Earth by inductive means. This canonical representation, not available yet in the literature to this time (beyond the static term), may apply to other practical cases than this one in geoelectromagnetics, whilst it adds useful reference re- sults to the already ample library of scattering by simple shapes using analytical methods.

Probability and spin densities of rare earth tri-positive ions

Probability and spin densities for rare-earth tri-positive ions in their ground states are calculated using irreducible tensor operators and Racah double tensors, which enable an expansion of the densities in (vector) spherical harmonics. Graphical representation of such densities reveal characteristic features lost in schematic shapes predicted by considering only the sign of the electrostatic quadruple moments. As a rule, due to the spin–orbit coupling, the spin density is not collinear. The only exceptions are found in the fully aligned ( M=± J) states of the heavy rare earths, where J= L+ S and S( r)∥ z. The implications of spin-density patterns for the nature of anisotropic exchange interactions are discussed.

Euler angle geometry, helicity basis vectors, and the Wigner D-function addition theorem

Euler angles often are used to describe the orientation of a secondary system rotated relative to a primary system by generating two auxiliary systems and the secondary with three rotations. We use basis vectors to show that the auxiliary systems of the y-convention Euler angles (used in quantum mechanics) are the cylindrical and spherical systems associated with the primary Cartesian system. We then apply Euler angle geometry to obtain an improved expression (and understanding) of the Wigner D-function addition theorem and related geometrical issues. We introduce helicity vectors and express vector spherical harmonics as combinations of Wigner D-functions to show that the D-function addition theorem is implicit within expansions of rotated vector spherical harmonics. As example applications of the Wigner D-function, we obtain an addition theorem for vector spherical harmonics and then simplify a dyadic Green function (related to Mie theory) describing inelastic light scattering from a laser-irradiated spherical droplet containing a fluorescing molecule.

Spectral and multiscale signal-to-noise thresholding of spherical vector fields

The basic concepts of spectral and multiscale selective reconstruction of (geophysically relevant) vector fields on the sphere from error-affected data is outlined in detail. The reconstruction mechanism is formulated under the assumption that spectral as well as multiscale approximation is well-representable in terms of only a certain number of expansion coefficients at the various resolution levels. It is shown that spectral denoising by means of orthogonal expansions in terms of vector spherical harmonics reflects global a priori information of the noise (e.g., in form of a covariance tensor field), whereas multiscale signal-to-noise thresholding can be performed under locally dependent noise information within a multiresolution analysis in terms of spherical vector wavelets. An application of the multiscale formalism to Earth's magnetic field determination is presented.

Dynamics of spherically symmetric vortex of laser radiation in a nonlinear medium

The possibility of realization of a spherically symmetric vortex of the electromagnetic field in a homogeneous isotropic medium is analyzed. It was shown that the vortex could emerge by the interaction of the vector spherical harmonic with the nonlinear medium having a negative linear permittivity. As a result the nonlinear refractive index becomes the valid value in a small volume near the centre. The radial-gradient distribution of the nonlinear refractive index acts as a spherical microcavity induced in the nonlinear medium. Within this volume a set of undamped vector spherical harmonics forms the spherical symmetric structure of the electromagnetic field of a soliton type. This structural resonance has quite discrete energy values. Mechanisms and conditions of self-organization of the spherically symmetric vortex are discussed on a basis of nonlinear-optical and quantum-mechanical approaches. Dynamics and stability of the spherically symmetric vortex in the nonlinear medium are discussed.

Quasi-static motion of a capillary drop, II: the three-dimensional case

A theory is presented to analyze the nonlinear stability of a drop of incompressible viscous fluid with negligible inertia. The theory is developed here on the three-dimensional version of the relevant free-boundary model for Stokes equations. Within this context we show that if the free-boundary initiates close to a sphere r=1+ελ 0(ω), |ε| small, ω=(θ,ϕ), then there exists a global-in-time solution with free boundary r=1+λ(ω,t,ε)=1+ ∑ n⩾1 λ n(ω,t)ε n, which approaches a sphere exponentially fast as t→∞. Moreover, we prove that if λ 0( ω) is analytic (resp. C ∞) in ω, then the velocity u → (x,t,ε) , the pressure p( x, t, ε) and the free-boundary λ are all jointly analytic (resp. C ∞) in ( x, ε). In an earlier paper, we considered the analogous problem for a two-dimensional drop. Although the three-dimensional problem proceeds along similar lines, the analysis is more complicated due to the fact that we work here with spherical harmonics and vector spherical harmonics.