Anisotropic Effects in Geometrically Isotropic Lattices

For spacings and element dimensions small with respect to wavelength the directric constant of a completely general lattice of identical elements may be represented by a tensor (k_e). In many applications of artificial dielectrics it is important that the dielectric act as an isotropic medium for microwaves. This requires that (k_e) reduce to a scalar. The dielectric constant tensor will reduce to a scalar only if the lattice is cubical and the geometry and the material of the elements is restricted so that the induced fields may be represented by a set of three mutually perpendicular static dipoles at the lattice points. Isotropy further requires that the moment of the resultant dipoles be proportional to the inducing field and that the proportionality factor be 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. This paper evaluates the anisotropy produced by an arbitrary ratio of element spacing to wavelength, and demonstrates that there is a basic anisotropy associated with the granularity of an array composed of isotropic elements arranged in structurally isotropic patterns.