Kristopher T. Kim

The translation formula for vector multipole fields and the recurrence relations of the translation coefficients of scalar and vector multipole fields

We present simple alternate derivations of the translation formula for spherical vector multipole fields and the recurrence relations satisfied by the translation coefficients of spherical scalar and vector multipole fields. The derivations use the well-known and widely used spherical tensor technique to conveniently couple and decouple quantities associated with spherical angular variables, and these coupling and decoupling of angular quantities substantially simplify the algebra involved. As a consequence, the derivations are quite concise, and the resulting expressions for the recurrence relations are more compact and general than the existing ones.

SYMMETRY RELATIONS OF THE TRANSLATION COEFFICIENTS OF THE SPHERICAL SCALAR AND VECTOR MULTIPOLE FIELDS

We offer symmetry relations of the translation coefficients of the spherical scalar and vector multi-pole fields. These relations reduce the computational cost of evaluating and storing the translation coefficients and can be used to check the accuracy of their computed values. The symmetry relations investigated herein include not only those considered earlier for real wavenumbers by Peterson and Strom [9], but also the respective symmetries that arise when the translation vector is reflected about the xy-, yz-, and zx-planes. In addition, the symmetry relations presented in this paper are valid for complex wavenumbers and are given in a form suitable for exploitation in numerical applications.

Truncation-Error Reduction in 2D Cylindrical/Spherical Near-Field Scanning

We introduce a near-field to far-field transformation for two-dimensional cylindrical/spherical scanning that significantly reduces angular-truncation errors. After examining the limitations of the traditional multipole-based expansion of truncated scan data, we consider an alternative expansion based on Slepian functions and show how far-field values can be extracted from the resulting expansion coefficients. We compare the performance and computational cost of the new transformation with those of the traditional one.

A memory-reduction scheme for the FFT T-matrix method

A method is presented that reduces the storage requirement of the FFT T-matrix method. It is based on the configuration- and Fourier-domain symmetry relations of the translation coefficients of the transverse spherical multipole fields. Only a minimum set of these matrices needs to be computed and stored. Elements of the full matrices for a given modal combination are rapidly generated using appropriate symmetry relations as the convolution is carried out. The cost of generating the full matrices from the minimum set scales as O(N), while the cost of performing a convolution grows as O(N logN). Thus, the presented memory-reduction scheme increases the CPU time only negligibly. In addition, the method can readily be adapted to an exiting FFT T-matrix code, enabling it to handle larger problems.

Direct Determination of the T-Matrix From a MoM Impedance Matrix Computed Using the Rao-Wilton-Glisson Basis Function

We present an explicit and numerically exact method for determining the T matrix of an arbitrarily shaped PEC object directly from a MoM impedance matrix that is generated with the Rao-Wilton-Glisson basis function for use in multiple-scattering calculations.

Homogenization of three-dimensional metamaterial objects and validation by a fast surface-integral equation solver.

A homogenization model is applied to describe the wave interaction with finite three-dimensional metamaterial objects composed of periodic arrays of magnetodielectric spheres and is validated with full-wave numerical simulations. The homogenization is based on a dipolar model of the inclusions, which is shown to hold even in the case of densely packed arrays once weak forms of spatial dispersion and the full dynamic array coupling are taken into account. The numerical simulations are based on a fast surface-integral equation solver that enables the analysis of scattering from complex piecewise homogeneous objects. We validate the homogenization model by considering electrically large disk- and cube-shaped arrays and quantify the accuracy of the transition from an array of spheres to a homogeneous object as a function of the array size. Simulation results show that the fields scattered from large arrays with up to one thousand spheres and equivalent homogeneous objects agree well, not only far away from the arrays but also near them.

Slepian transverse vector spherical harmonics and their application to near-field scanning

We introduce the Slepian transverse vector spherical harmonics (TVSH). Unlike the classical TVSH, they are orthogonal over a given truncated spherical surface and the orthogonality constants can be computed. We apply the Slepian TVSH to the problem of reconstructing the far field from spatially truncated near-field samples and show that the far field can be reconstructed accurately over the entire angular sector where the near-field samples are available.

Efficient Recursive Generation of the Scalar Spherical Multipole Translation Matrix

New efficient recursive procedures for generating the translation matrix of the scalar spherical multipole field are described. They are based on a new set of recurrence relations that result when the angular-momentum operator is applied to the spherical multipole field. Their efficiency and accuracy are compared analytically and through a computer experiment with those of the brute-force method and an existing recursive procedure.

On the computational efficiencies of two FFT-based solution methods for the scalar multiple-scattering equation

We compare and improve the computational efficiencies of two FFT-based solution techniques for the scalar t-matrix multiple-scattering equation. The FFT operation requires field quantities be expressed on a regular grid and the two techniques differ in how to go about achieving this; the first technique uses the non-diagonal translation operator of the spherical multipole field, while the second method uses the diagonal translation operator of Rokhlin (1998). Because of its use of the non-diagonal operator, the first technique has been thought to require a greater number of FFT operations than the second one. We first show that both techniques require about the same number of FFT operations. After considering the symmetry relations of the non-diagonal and diagonal translation operators, we show that these symmetry relations can be used to reduce the respective memory requirements of the two FFT t-matrix methods

Truncation-Error Reduction in Spherical Near-Field Scanning Using Slepian Sequences: Formulation for Scalar Waves

We discuss the error that results when the far field is reconstructed from spatially truncated near-field samples and present an effective mitigation technique based on the Slepian sequence for acoustic spherical near-field scanning. We show that the truncation error is inevitable whenever the far field is reconstructed using the classical near-field-to-far-field transformation. After discussing the Slepian sequence for a truncated spherical surface and its analytic and numerical properties, we apply it to expand truncated NF samples and derive the near-field-to-far-field transformation of the resulting expansion coefficients, from which the far field can be computed. We demonstrate the efficacy of this transformation by applying it to near-field scanning for bistatic scattering from a sphere and radiation from a current distribution.