This talk concerns composite materials that have periodicity in a plane. They are formed by identical, long cylinders that constitute a two-dimensional, arbitrary Bravais lattice in the plane perpendicular to the cylinders. The cross-section of the cylinders is also of arbitrary form. The cylinder material and the intersticial material are assumed to be homogeneous and isotropic. In the last decade, such structures came to be known as photonic orphononic crystals - depending whether the materials involved are of transparent or of elastic nature. The study of these periodic composites usually focuses on a high-frequency band gap - the phononic or photonic gap in which, respectively, light or sound cannot propagate. Here, on the contrary, we are interested in the low-frequency regime, where the wavelength (corresponding to the Bloch vector) is much greater than the lattice spacing. Thus, we present a theory of homogenization applicable to a very general class of composites with two-dimensional periodicity.[1] Our method is applicable to a wide class of materials: dielectric, magnetic, electrically conducting, thermally conducting, etc. For simplicity, the material properties are given in terms of the dielectric constants of the cylinder and host substances.
The homogenization is performed by taking the quasistatic limit of Maxwell's equations. In this limit, we prove that the composite behaves like either a uniaxial or a biaxial (homogeneous)optical medium. That is, it can be described by means of a hermitian dielectric tensor that can be diagonalized. It has three eigenvalues -the Principal Dielectric Constants- that, in general, are all different. We have derived exact and compact formulas for these dielectric constants, given in terms of summations over the two-dimensional reciprocal lattice and a matrix inversion.[1] These formulas have been applied to a variety of Bravais lattices, cylinder cross-sectional shapes, and materials.[1,2] In each case, the Principal Dielectric Constants are computed as a function of the cylinder filling fraction. Convergence is very rapid, and some of our results are of unprecedented precision.
[1] P.Halevi, A.A.Krokhin, and J.Arriaga, Phys.Rev.Lett.82,719(1999)
[2] P.Halevi, A.A.Krokhin, and J.Arriaga, Appl.Phys.Lett.75,2725(1999)