Anisotropic Effects in Geometrically Isotropic Lattices

Abstract

The study of the dielectric properties of a lattice composed of identical metallic or dielectric elements of various geometries has re- ceived considerable attention in recent years in connection with the prac- tical application of such structures for polarizing devices, microwave lenses, and radome materials. It has been shown by the author1 that, for spacings and element dimensions small with respect to wavelength, the dielectric constant of a completely general uniform lattice of identical elements may be represented by a tensor (k_e) which may be written in functional form as (k_e) = f[(T), (δ)], where (T) is the structural anisotropy tensor which is related to the geometry of the lattice and (δ) is the polarizability tensor of the elements of the lattice. (δ) describes element anisotropy which may be due to material, the shape of the element, or both. An example of a lattice element displaying only material anisotropy is a ferrite or gaseous element of spherical shape immersed in a magnetostatic field. An example of shape anisotropy is the case of metallic or dielectric objects of non-spherical shape. In general, then, for spacing and element dimensions small compared to wavelength, there can be three orders of anisotropy in a lattice—structural or lattice anisotropy, material anisotropy, and anisotropy because of element shape. Examples of each type are described and discussed in detail in the aforementioned reference. A fourth order of anisotropy, which is related to the granularity of the lattice and be- comes important at higher frequencies will be described and investigated in this paper. It is usually important that the artificial material be as close as possible in physical properties to a real dielectric. This requires that it should be isotropic; that is, the structure should exhibit the same properties for a plane wave propagating through it in any direction. This behavior requires that (k_e) reduce to a scalar which, in turn, demands that (T) and (δ) become scalars, except for the special case in which the structural anisotropy of the array is compensated for by the element anisotropy in which case (k_e) is reduced to a scalar. The structural anisotropy tensor (T) will reduce to a scalar only if the lattice is cubical, while (δ) becomes a scalar only if the geometry and material of the lattice elements are so restricted that the induced fields may be represented by a set of three mutually perpendicular static dipoles on the lattice points.1 Isotropic behavior further requires that the moments of the resultant dipoles must be proportional to the inducing field. The proportionality factor is a scalar independent of direction. However, at shorter wavelengths the representation of the lattice elements by static dipoles will not be valid and the medium becomes anisotropic. The main objective of this paper is to evaluate the anisotropy produced by the finite ratio of wavelength to element spacing and to show that the Clausius-Mosotti relation so often used in predicting the properties of artificial lattice dielectrics is a satisfactory approximation only if the spacing is very small with respect to wavelength.

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