Traveling Waves on Two- and Three-Dimensional Periodic Arrays of Lossless Acoustic Monopoles, Electric Dipoles, and Magnetodielectric Spheres

Abstract

Abstract : Traveling waves on 2D and 3D infinite periodic arrays of small lossless acoustic monopoles, electric dipoles, and magnetodielectric spheres are investigated. The waves are assumed to propagate along the axis of the arrays. The focus is on obtaining the kd-BETAd equations (diagrams) characterizing the traveling waves. Initial forms of the kd-BETAd equations are obtained by summing the fields scattered from all the array elements incident on a reference element of the array located at the origin of the coordinate system. These initial forms of the kd-BETAd equations are, however, useless for computational purposes since the summations converge extremely slowly. The core of the investigation is, therefore, to convert the slowly convergent summations to forms that can be used for efficient calculation of the kd-BETAd curves. These conversions are performed by first using either the Poisson summation formula or a method based on the use of Floquet mode expansions. Expressions for the efficient summation of Schloemilch series are then used to obtain the final forms of the kd-BETAd equations. Exact computable expressions for the fields of 3D acoustic monopole, electric dipole, and 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-BETAd curves for the corresponding infinite arrays.