Algorithm 1018: FaVeST—Fast Vector Spherical Harmonic Transforms

Vector spherical harmonics on the unit sphere of ℝ 3 have broad applications in geophysics, quantum mechanics, and astrophysics. In the representation of a tangent vector field, one needs to evaluate the expansion and the Fourier coefficients of vector spherical harmonics. In this article, we develop fast algorithms (FaVeST) for vector spherical harmonic transforms on these evaluations. The forward FaVeST evaluates the Fourier coefficients and has a computational cost proportional to N log √ N for N number of evaluation points. The adjoint FaVeST, which evaluates a linear combination of vector spherical harmonics with a degree up to ⊡ M for M evaluation points, has cost proportional to M log √ M . Numerical examples of simulated tangent fields illustrate the accuracy, efficiency, and stability of FaVeST.

Analysis of Light Scattering Properties about Dent Nanoparticles upon Wafer

The light scattering properties of the dent nanoparticles upon wafers is discussed in this paper. Taking the advantage of the Bobbert-Vlieger (BV) theorem, the scattering model between wafer and dent nanoparticles is established. The scattering process is analyzed and the scattering coefficients are derived by using of the vector spherical harmonic function. The differential scattering cross section (DSCS) of the dent nanoparticles upon the wafer is calculated which is compared with the extended Mie method proved the validity of the method and the influences of the dent position, dent scale and scattering angle on the DSCS are analyzed numerically in details. The result is shown that the effect of the dielectric is smaller than the metal. Therefore, the material of the defect and the shape can be extracted by calculate the DSCS, which provide strong theoretical foundation to the nondestructive detector engineer.

Direct Simulation of Scattering and Absorption by Particle Deposits

A computational scheme is presented to exactly calculate the electromagnetic field distribution, and associated radiative absorption and scattering characteristics, of large-scale ensembles of spherical particles that are subjected to a focussed incident beam. The method employs a superposition extension to Lorenz/Mie theory, in which the internal and scattered fields for each sphere in the ensemble are represented by vector spherical harmonic expansions, and boundary conditions at the surfaces of the spheres are matched by application of the addition theorem for vector harmonics. The incident field is modeled as a transverse, linearly-polarized wave with a Gaussian amplitude distribution along a fixed focal plane. Application of the method to prediction of the absorption and reflectance characteristics of particle deposits is discussed, and illustrative calculations are presented.

Analysis of radiative scattering for multiple sphere configurations

An analysis of radiative scattering for an arbitrary configuration of neighbouring spheres is presented. The analysis builds upon the previously developed superposition solution, in which the scattered field is expressed as a superposition of vector spherical harmonic expansions written about each sphere in the ensemble. The addition theorems for vector spherical harmonics, which transform harmonics from one coordinate system into another, are rederived, and simple recurrence relations for the addition coefficients are developed. The relations allow for a very efficient implementation of the ‘order of scattering’ solution technique for determining the scattered field coefficients for each sphere.