Three-Dimensional Lattices with Isotropic Dielectric Properties

Expressions are derived for the constitutive dielectric parameters of a cubical lattice whose elements consist of a triad of mutually perpendicular polarizable elements. The analysis gives the fundamental relations for the simulation by suitably disposed dipoles, of three-dimensionally isotropic dielectrics with dielectric constants greater than, equal to, and less than unity. Three different approaches have been used. One of these is a complete and rigorous solution which gives the dielectric tensor for the general case of unrestricted spacing to wavelength ratio. This rigorous analysis shows that the Clausius-Mosotti relation often used in predicting the properties of such lattices is a satisfactory approximation only if the spacing is very small with respect to wavelength. Using the general principles developed in the paper, conditions are derived for the realizability of reflectionless media.