Vector Spherical Harmonics Research Articles

Theory of complete orthonormal relativistic vector wave function sets and Slater type relativistic vector orbitals in coordinate, momentum and four-dimensional spaces

The new analytical relations for the relativistic vector wave functions and Slater type relativistic vector orbitals in coordinate, momentum and four-dimensional spaces are derived using the properties of quasirelativistic vector spherical harmonics introduced by the author in previous paper (I.I. Guseinov, J. Math. Chem., 44, 197 (2008)) and complete orthonormal scalar basis sets of nonrelativistic ψ α -exponential type orbitals (ψ α -ETO), \({\phi ^{\alpha}}\)-momentum space orbitals (\({\phi ^{\alpha}}\)-MSO) and z α -hyperspherical harmonics (z α -HSH). The 6-component relativistic vector wave functions obtained are complete without the inclusion of the continuum. The relativistic vector wave function sets and Slater type relativistic vector orbitals are expressed through the corresponding quasirelativistic vector wave functions and Slater vector orbitals, respectively. The analytical formulas are also derived for overlap integrals over Slater type relativistic vector orbitals with the same screening constants in coordinate space.

Optical response of metal or dielectric nano-objects in strongly convergent light beams

We present a general formalism for describing the optical response of spherical nano-objects located in the focus region of a tightly convergent light beam. The theory can be applied to any polarized light beam emerging from axially symmetric focusing optical devices. The approach is based on the generalized Mie theory equations and the expansion of the angular spectrum plane-wave decomposition of the beam in terms of vector spherical harmonics. The numerical efficiency of the formalism is exemplified in the case of reflecting microscope objectives, in the context of single-particle experiments based on the spatial modulation spectroscopy technique, for spherical gold spheres and pairs of nearly touching gold particles.

An efficient hybrid MLFMA–FFT solver for the volume integral equation in case of sparse 3D inhomogeneous dielectric scatterers

Electromagnetic scattering problems involving inhomogeneous objects can be numerically solved by applying a Method of Moments discretization to the volume integral equation. For electrically large problems, the iterative solution of the resulting linear system is expensive, both computationally and in memory use. In this paper, a hybrid MLFMA–FFT method is presented, which combines the fast Fourier transform (FFT) method and the High Frequency Multilevel Fast Multipole Algorithm (MLFMA) in order to reduce the cost of the matrix–vector multiplications needed in the iterative solver. The method represents the scatterers within a set of possibly disjoint identical cubic subdomains, which are meshed using a uniform cubic grid. This specific mesh allows for the application of FFTs to calculate the near interactions in the MLFMA and reduces the memory cost considerably, since the aggregation and disaggregation matrices of the MLFMA can be reused. Additional improvements to the general MLFMA framework, such as an extention of the FFT interpolation scheme of Sarvas et al. from the scalar to the vectorial case in combination with a more economical representation of the radiation patterns on the lowest level in vector spherical harmonics, are proposed and the choice of the subdomain size is discussed. The hybrid method performs better in terms of speed and memory use on large sparse configurations than both the FFT method and the HF MLFMA separately and it has lower memory requirements on general large problems. This is illustrated on a number of representative numerical test cases.

Photon angular momentum: selection rules and multipolar transition moments

The representation of electromagnetic radiation by means of vector spherical harmonics and Bessel functions is described and used to derive selection rules for atomic electronic transitions which, as expected, show that the angular momentum of the system—atom plus photon—is conserved and, more importantly, provide a very precise description of the angular momentum of the photon involved. Examples of matrix elements for electric dipole, electric quadrupole and magnetic dipole transition moments of the hydrogen atom are calculated and the results compared with those obtained by the usual plane-wave expansion and with more recent results obtained in connection with studies of ‘twisted’ beams.

The Responses of the Deformable Earth to Different Driving Forces

The spatial and temporal variations of the Earth deformation and the gravitational field are important both in the theoretical research and in the construction of geospatial database. The Earth deforms due to various mechanisms and the deformation further induces changes in the gravitational potential of the Earth, i.e. the deformation-induced additional potential or the Euler gravitational increment. Based on the theory of vector spherical harmonics, we discuss in this paper the Earth deformation and gravitational increment resulting from the tidal force, loading force and the stress of the Earth's surface. We write out the expression for the Euler gravitational potential increment and the relations between different Love numbers. These are all important points in the research on Earth deformation.

Description of Multipole inf-Electron Systems

A systematic description of multipole degrees of freedom is discussed on the basis of the Stevens' operator-equivalent technique. The generalized Stevens' multiplicative factors are derived for all of the electric and the magnetic multipoles relevant to f-electron systems. With extensive use of the Stevens' factors, we express the spatial dependences of the electric and the magnetic fields, and the electric and the magnetic charge densities of localized f electrons. The latter is utilized to draw wave functions including their magnetic profile in addition to their shape with the charge density. The definite relation between the operators as quantum-mechanical variables in a multipole exchange model and the multipole moments in expansion of electromagnetic fields is given. The general treatments for the exchange model with the RPA susceptibility and the Ginzburg-Landau free-energy expansion are discussed, using CexLa1-xB6 as a typical example. The representative formula of the vector spherical harmonics are summarized, which are suitable basis for vector fields in the spherical expansion.

Matrix approach for light scattering from a multilayered rotationally symmetric bianisotropic sphere

Electromagnetic wave propagation in rotationally symmetric bianisotropic structures is investigated. Spherical solutions are presented in compact tensor (matrix) form with separated angle and radial variables. The separation is achieved by applying tensor functions ${F}_{lm}(\ensuremath{\theta},\ensuremath{\varphi})$ based on ordinary scalar and vector spherical harmonics. Such tensor functions describe the angle dependences of partial spherical waves. The scattering problem for a multilayered bianisotropic spherical particle is solved using an impedance tensor and evolution operator (transfer matrix). These quantities have been previously defined for planar and cylindrical structures. The approach developed is applied to computing the differential cross section of a plane wave from a two-layer bianisotropic sphere.

Solving the diffraction problem of electromagnetic waves on objects with a complex geometry by the pattern equations method

A new efficient method for solving the problems of waves diffraction on complex-shaped objects with the use of their replacement by a group of bodies with more simple form (fragments of complex objects) is offered. By the expansion of the scattering patterns of separate bodies in the series of vector spherical harmonics, the problem is reduced to solving the algebraic system of equations. It is shown that the method possesses a high convergence rate. Examples of modeling the scattering patterns of various complex objects are considered. Reliability of the results obtained is validated using the Optical theorem.

Optical response of a single spherical particle in a tightly focused light beam: application to the spatial modulation spectroscopy technique

We develop a new and numerically efficient formalism to describe the general problem of the scattering and absorption of light by a spherical metal or dielectric particle illuminated by a tightly focused beam. The theory is based on (i) the generalized Mie theory equations, (ii) the plane-wave decomposition of the converging light beam, and (iii) the expansion of a plane wave in terms of vector spherical harmonics. The predictions of the model are illustrated in the case of silver nanoparticles. The results are compared with the Mie theory in the local approximation. Finally, some effects related to the convergence of the beam are analyzed in the context of experiments based on the spatial modulation spectroscopy technique.

On the complementarity of electroencephalography and magnetoencephalography

We show that for the spherical model of the brain, the part of the neuronal current that generates the electric potential (and therefore the electric field) lives in the orthogonal complement of the part of the current that generates the magnetic potential (and therefore the magnetic induction field). This means that for a continuously distributed neuronal current, information missing in the electroencephalographic data is precisely information that is available in the magnetoencephalographic data, and vice versa. In this way, the notion of complementarity between the imaging techniques of electroencephalography and magnetoencephalography is mathematically defined. Using this notion of complementarity and expanding the neuronal current in terms of vector spherical harmonics, which by definition provide the angular dependence of the current, we show that if the electric and the magnetic potentials in the exterior of the head are given, then we can determine certain moments of the functions which provide the radial dependence of the neuronal current. In addition to the above notion of complementarity, we also present a notion of unification of electroencephalography and magnetoencephalography by showing that they are governed respectively by the scalar and the vector invariants of a unified dyadic field describing electromagnetoencephalography.

The problem of electromagnetic-wave diffraction from a group of bodies solved via the pattern-equation method

An efficient method is proposed for solving the problem of electromagnetic-wave diffraction from a group of objects. Scattering patterns are decomposed in vector spherical harmonics to reduce the problem to a system of algebraic equations. It is shown that the method exhibits a high convergence rate. The scattering patterns of certain single objects are compared to the scattering patterns of the bodies formed from halves of these objects. The validity of the optical theorem is checked.

The Local Stellar Velocity Field via Vector Spherical Harmonics

We analyze the local field of stellar tangential velocities for a sample of $42 339$ non-binary Hipparcos stars with accurate parallaxes, using a vector spherical harmonic formalism. We derive simple relations between the parameters of the classical linear model (Ogorodnikov-Milne) of the local systemic field and low-degree terms of the general vector harmonic decomposition. Taking advantage of these relationships we determine the solar velocity with respect to the local stars of $(V_X,V_Y,V_Z)=(10.5, 18.5, 7.3)\pm 0.1$ \kms. The Oort's parameters determined by a straightforward least-squares adjustment in vector spherical harmonics, are $A=14.0\pm 1.4$, $B=-13.1\pm 1.2$, $K=1.1\pm 1.8$, and $C=-2.9\pm 1.4$ \kmspc. We find a few statistically significant higher degree harmonic terms, which do not correspond to any parameters in the classical linear model. One of them, a third-degree electric harmonic, is tentatively explained as the response to a negative linear gradient of rotation velocity with distance from the Galactic plane, which we estimate at $\sim -20$ \kmspc. The most unexpected and unexplained term within the Ogorodnikov-Milne model is the first-degree magnetic harmonic representing a rigid rotation of the stellar field about the axis $-Y$ pointing opposite to the direction of rotation. This harmonic comes out with a statistically robust coefficient $6.2 \pm 0.9$ \kmspc, and is also present in the velocity field of more distant stars. The ensuing upward vertical motion of stars in the general direction of the Galactic center and the downward motion in the anticenter direction are opposite to the vector field expected from the stationary Galactic warp model.

Evaluation of Signal Space Separation via simulation

Abstract Signal Space Separation (SSS) method is based on the solution of Laplace's equation: expansion of the spherical harmonics. It has been realized and proved that the series of vector spherical harmonics solution can be split into two parts that are related to the MEG (magnetoencephalography) measurements: brain signals and external interferences. In practice, SSS has become an elegant and promising method to recover bio-magnetic signal and remove external disturbance in empirical MEG measurements. In the present study, we evaluate SSS comprehensively via computer simulation and real applications. We generate two types of interference sources: magnetic dipole and electric current dipole. We examine the suppression effect for interference signals, and the goodness of the reconstruction for the interested signal. The expansion order of vector spherical harmonics plays critical role in both signal reconstruction in the brain and external artifact reduction, this is studied thoroughly and demonstrated in this paper. Besides computer simulation, SSS method is also applied on real MEG data sets and the effectiveness is investigated. Finally, we give an objective conclusion of the advantage and limitation of SSS method, and assess its value in practical MEG measurements.

Polynomial bases for subspaces of vector fields in the unit ball. Method of ridge functions

This paper deals with the construction of orthogonal polynomial bases for particular subspaces of vector fields defined in the unit ball of ℝ3. Our approach uses vector spherical harmonics to construct orthogonal sets of specific solenoidal and potential vector fields by means of ridge functions. It is shown that the approach leads to bases according to the subspaces induced by the Helmholtz–Hodge decomposition of square integrable vector fields.

Symmetry transformation matrices for structures

Many structures in nature and engineering are symmetric. Depending on the degree of symmetry, it is possible to simplify the computations considerably by block diagonalizing the stiffness matrices. Closed-form solutions of transformation matrices for such block diagonalizations can be derived using group theory for arbitrary symmetry groups. This paper presents closed-form solutions of transformation matrices based on an alternative derivation. It is shown that transformation matrices for C nv and D nh groups can be obtained from a finite Fourier series decomposition of load and displacement vectors. Furthermore, it is shown that structures with tetrahedral, octahedral and icosahedral symmetries can be block diagonalized in an elegant way using vector spherical harmonics.

Vector spherical harmonics application to 3-D tomography problem

A method of series expansion with the aid of vector spherical harmonics intended for inverting line integrated data is proposed to investigate 3-D vector fields in the spherical plasmas. A set of numerical computations demonstrating the 3-D reconstruction of the model vector fields has been performed to assess the inversion method proposed.

Reply to “Comment on ‘Infrared and Raman selection rules for elastic vibrations of spherical nanoparticles' ”

In this Reply, it is shown that, contrary to Goupalov, Saviot, and Duval's argument, apparently contradictory parity assignments for the vector spherical harmonics ${\mathbf{Y}}_{l{l}^{\ensuremath{'}}m}$ by different authors, when applied consistently, all yield our selection rule.

Separating inner and outer Earth's magnetic field from CHAMP satellite measurements by means of vector scaling functions and wavelets

SUMMARY This article is concerned with an approach of modelling the Earth's magnetic field, as measured by satellites, in terms of vector kernel functions, which we will call vector scaling functions and wavelets. The major ingredient is the presentation of a system of vector spherical harmonics which separates a given spherical vector field with respect to its sources, that is, the spherical vector field is separated into a part which is generated by sources inside the sphere, a part which is generated by sources outside the sphere and a part which is generated by sources on the sphere, which are, for example, currents crossing the sphere. Using this special system of vector spherical harmonics vector scaling functions and wavelets are constructed which keep the advantageous property of separating with respect to source but which also allow a locally adapted modelling of the respective vector field. At the end of the article, the method is tested on magnetic field data measured by the German geoscientific research satellite CHAMP.

Wavelet Decomposition of Spherical Vector Fields with Respect to Sources

This article is concerned with an approach of modelling the Earth’s magnetic field as measured by satellites in terms of a special system of vector spherical harmonics and in terms of vector kernel functions, called vector scaling functions and wavelets. The main ingredient is the presentation of a system of vector spherical harmonics which separates a given spherical vector field with respect to its sources, i.e., the spherical vector field is separated into a part which is induced by sources inside the sphere, a part which is induced by sources outside the sphere and a part which is induced by sources on the sphere, which are, for example, currents crossing the sphere. Using this special system of vector spherical harmonics vector scaling functions and wavelets are constructed which keep the advantageous property of separating with respect to sources but which also allow a locally reflected modelling of the respective vector field. At the end of the article, the method is tested on real magnetic field data measured by the German geoscientific research satellite CHAMP.

Mie scattering by an anisotropic object Part II Arbitrary-shaped object: differential theory

The differential theory of diffraction by an arbitrary-shaped body made of arbitrary anisotropic material is developed. The electromagnetic field is expanded on the basis of vector spherical harmonics, and the Maxwell equations in spherical coordinates are reduced to a first-order differential set. When discontinuities of permittivity exist, we apply the fast numerical factorization to find the link between the electric field vector and the vector of electric induction, developed in a truncated basis. The diffraction problem is reduced to a boundary-value problem by using a shooting method combined with the S-matrix propagation algorithm, formulated for the field components instead of the amplitudes.