Traveling waves on two‐ and three‐dimensional periodic arrays of lossless scatterers

Abstract

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.