An analysis is presented of scattering of an electromagnetic linearly polarized plane wave by a multilayered sphere. The focus is on obtaining a computational form of the Mie coefficients for the scattered field. A central role is played by ratios of spherical Bessel functions that can be calculated easily, rapidly, and accurately by recurrence relations whose stabilities are demonstrated. Logarithmic derivatives are not employed. A detailed outline is given of a carefully tested computer program for implementing and validating the analysis. Numerous comparisons are given of numerical results obtained with this program with corresponding results in the literature. Important properties of the Mie coefficients and aspects of the scattered field are discussed including the loci of the Mie coefficients in the complex plane; the resonances of the Mie coefficients; the extinction, scattering, and absorption efficiencies of the scattered field; radiation pressure; the Debye series, and the complex angular momentum (CAM) method.

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.

This is the first part of a two‐part series dealing with complex dipolar waves propagating along the axes of 1D, 2D, and 3D infinite periodic arrays of small lossless and lossy permeable spheres. The theory is presented in this paper and numerical results are presented by Shore and Yaghjian (2012). The focus is on the dispersion (k–β) equations relating the array propagation constant, β, to the free‐space wave number,k, for dipolar complex waves. The k–β equation for the complex propagation constants of a given array is obtained from the corresponding equation previously obtained for the real propagation constants by rewriting the real propagation dispersion equation in a form that can be analytically continued into the complex β plane. This equation reduces correctly to the real β dispersion equation and enables complex values of β to be found as a function of the array element parameters. By allowing for all the possible branches of the multivalued homogeneous dispersion equation analytically continued into the complex βplane, the propagation constants of all the improper as well as proper complex waves supported by the 1D, 2D, and 3D arrays are found from the homogeneous solutions for these arrays. Green's functions for external sources are not required to find the propagation constants of the complex waves supported by the arrays. For 3D arrays, in certain frequency ranges, it is possible to regard the arrays as media characterized by bulk or effective permittivities and permeabilities. Expressions for these bulk parameters, more accurate than the Clausius‐Mossotti expressions, are obtained from quantities readily available in the solutions of the dispersion equations.

This is the second part of a two‐part series dealing with complex dipolar waves propagating along the axes of 1D, 2D and 3D infinite periodic arrays of small lossless and lossy permeable spheres. Shore and Yaghjian (2012) provide the theory of the complex waves and the dispersion (k–β) equations for their propagation constants. In this paper we present and discuss representative dispersion diagrams for arrays of magnetodielectric spheres (spheres with appreciable permittivity and permeability), diamond spheres, and silver nanospheres.

Because the general properties of complex waves are not contained in any one reference of which we are aware, we summarize some of the general properties of complex waves supported by uniform or periodic source-free waveguides and discuss some of the distinguishing features of fast and slow waves.

In the above titled paper (ibid., vol. 57, no. 10, pp. 3077-3091, Oct. 2009), there were a number of errors. These errors are remedied here.

We investigate the causality relations in homogenized metamaterial arrays and determine the reasons why metamaterial effective (bulk) constitutive parameters obtained using classic point-dipole approximations may violate basic causality conditions represented by the Kramers-Kronig relations. We show that noncausality is inherently introduced by the use of point-dipole approximations within Maxwell-Garnett homogenization procedures and that these artifacts become particularly significant for more densely packed arrays. In contrast we show that properly defined and exactly computed bulk constitutive parameters of periodic metamaterial arrays always satisfy causality for a fixed spatial frequency in a rigorous homogenization framework. Finally we show how the Maxwell-Garnett approach can be modified to effectively remove the noncausal effects.

It is shown that any spatially and temporally dispersive bianisotropic material, satisfying Maxwell's macroscopic equations for E and H in the Fourier transformed (β, ω) space, can also be represented as an anisotropic material. Thus, for many applications, magnetoelectric constitutive parameters can be avoided at the macroscopic level.

The dispersion equation is derived and solved for the complex propagation constants of traveling waves (eigenmodes) supported by three-dimensional, infinite, periodic arrays of identical, electrically small, lossy or lossless, magnetodielectric spheres. Expressions are obtained for the effective permittivity and permeability of the arrays in frequency regions where the arrays can be regarded as continuous, homogeneous, isotropic media.

: Complex waves propagating along the axes of 1D, 2D and 3D infinite periodic arrays of small lossless and lossy magnetodielectric spheres are investigated. The focus is on obtaining the kd-BETAd equations (diagrams) characterizing dipolar complex waves. The kd-BETAd (dispersion) equation for the complex propagation constants of a given array is obtained from the corresponding equation previously obtained for the real propagation constants by rewriting the real propagation dispersion equation in a form that can be analytically continued into the complex BETAd plane. This equation reduces correctly to the real BETAd dispersion equation and enables complex values of BETAd to be found as a function of kd and the array element parameters. By allowing for all the possible branches of the multivalued homogeneous dispersion equation analytically continued into the complex BETAd plane, the propagation constants of all the improper as well as proper complex waves supported by the 1D, 2D, and 3D arrays are found from the homogeneous solutions for these arrays. Green's functions are not required to find the propagation constants of the complex waves supported by the arrays. The kd-BETA equations for complex BETAd are solved by searching a given region of the complex BETAd plane with a progressively finer grid for the value of BETAd that minimizes the absolute value of the dispersion equation. A final very accurate value for the zero is then obtained using an IMSL implementation of a quasi-Newton algorithm or of Mueller's Method. Computer programs have been written to obtain the kd-BETAd diagrams for all the arrays treated, and extensive numerical results are presented and discussed. For 3D arrays of magnetodielectric spheres, it is possible in certain regions of the kd-BETAd diagrams to regard the arrays as media characterized by bulk or effective permittivities and permeabilities.

A comprehensive investigation of traveling waves with complex propagation constants on 1D, 2D, and 3D infinite periodic arrays of electrically small lossless and lossy magnetodi-electric spheres is summarized. The focus is on obtaining and solving the dispersion equations for the propagation constants.

An exact k - beta (dispersion) equation to within the dipole scattering approximation has been obtained for a 3D array of two different alternating magnetodielectric spheres. The dispersion equation has the form of equating to zero the determinant of a system of four homogeneous equations in the normalized scattered field coefficients. Computationally efficient expressions are obtained for the coefficients of the homogeneous equation system as functions of the sphere radii, permittivities, and permeabilities, the free-space electrical separation distance of the array elements (kd), and the electrical separation distance (beta d) for the traveling wave supported by the array. For a given value of kd and an array of lossless scatterers the determinant equation can be solved for real beta d by a simple search procedure. For an array of lossy scatterers beta d is complex and a more difficult minimization in the complex plane is required to solve the determinant equation. The solution to the dispersion equation also yields values for the effective permittivity and permeability of the array regarded as a continuous medium. Computations were performed to investigate the performance of two-sphere arrays of lossless dielectric spheres, with the permittivities and radii of the two different dielectric spheres composing the array chosen so that the first magnetic dipole resonant frequency of one set of spheres equals the first electric dipole resonant frequency of the second set of spheres. Although it is shown that arrays composed of two different alternating purely dielectric spheres can behave as isotropic DNG media unlike arrays of identical dielectric spheres, the bandwidths are considerably narrower than those achievable with arrays of identical magnetodielectric spheres with appreciable permittivity and permeability close to each other. The practicality of using arrays of alternating two different purely dielectric spheres to fabricate DNG media depends on whether the narrow bandwidths are acceptable for the desired applications.

[1] In the paper “Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers” by R. A. Shore and A. D. Yaghjian (Radio Science, 42, RS6S21, doi:10.1029/2007RS003647, 2007), please note the following corrections. [2] Replace paragraphs 105–107 in section 5 with the single paragraph: [4] In Appendix B, replace −0.4902 in equation (B7c) by +0.4902. [5] In equation (B9) of Appendix B, ζ(3) = 1.20205⋯, where ζ is the Riemann zeta function.

An exact computable expression is obtained for the electromagnetic field of a three-dimensional partially finite periodic array of lossless or lossy magnetodielectric spheres illuminated by a plane wave propagating parallel to the array axis. The array is finite in the direction of the array axis and is of infinite extent in the directions transverse to the array axis. Illustrative numerical examples are presented.

The kd–βd (dispersion) equations are found for traveling waves on two‐ and three‐dimensional infinite periodic arrays of small lossless acoustic monopoles, electric or magnetic dipoles, and magnetodielectric spheres. Using Floquet mode expansions and then expressions for the rapid summation of Schlömilch series, prohibitively slowly convergent summations are converted to forms that can be used for the efficient calculation of the kd–βd equations. Computer programs have been written to obtain the kd–βd diagrams for all the arrays treated, and representative numerical results are presented and discussed. Expressions, more accurate than the Clausius‐Mossotti relations, are obtained for the effective or bulk permittivity and permeability of the arrays utilizing quantities readily available from the solutions of the kd–βd equations. Exact computable expressions for the fields of three‐dimensional lossless or lossy magnetodielectric sphere arrays that are finite in the direction of the array axis, illuminated by a plane wave parallel to the array axis, are obtained from the analyses performed to obtain the kd–βd curves for the infinite arrays.

This paper describes an analytic investigation of traveling waves on two-dimensional (2D) and three-dimensional (3D) infinite periodic arrays of lossless magnetodielectric spheres using a spherical-wave source scattering-matrix formulation. We limit our attention to traveling waves propagating in the direction of the array axis. It is assumed that only the lowest order multipole fields (electric and magnetic dipoles) are significant in analyzing scattering from the array elements. We focus on the kd-ßd equation (diagram) that relates the traveling wave electrical separation distance ßd of the array elements in the direction parallel to the array axis to the corresponding free-space electrical separation distance kd.

In Shore et al. (2004) we have employed the source scattering-matrix formulation with vector spherical wave functions to investigate traveling electromagnetic waves on infinite linear periodic arrays of lossless electric dipoles (which can be spheres that scatter as electric dipoles) and lossless magnetodielectric spheres (which can scatter as both electric and magnetic dipoles), respectively. It is assumed that either the spheres are sufficiently small that only the dipole scattered fields can be excited, or the frequency is such that all the scattered multipole fields are negligible except the dipole fields. Accordingly, the sphere scattering is treated using only electric and magnetic dipole vector spherical waves, the dipoles being orthogonal to each other and to the array axis. Although our investigation was motivated in part by the recent theoretical demonstration by Holloway et al. (2003) that a doubly negative medium can be formed by embedding an array of magnetodielectric spheres in a background matrix, the results are equally applicable to arrays of lossless "metallic" nanospheres with negative relative permittivity and relative permeability equal to one. Accordingly, the purpose of this paper is to summarize the analyses given in detail in Shore et al. (2004), and to use them to determine the k-β diagrams for the traveling waves on linear arrays of magnetodielectric spheres and metallic nanospheres.

A brief review is given of the derivation and application of dual-surface integral equations, which eliminate the spurious resonances from the solution to the original electric-field and magnetic-field integral equations applied to perfectly electrically conducting scatterers. Emphasis is placed on numerical solutions of the dual-surface electric-field integral equation for three-dimensional perfectly electrically conducting scatterers.