Efficient computation of the quasi-static effective permittivity for non-cubical periodic lattices

Artificial dielectrics are being studied in relation with the fabrication of lens antennas for shaped-beam applications. We present an efficient computational method for the calculation of the effective media parameters for non-cubical periodic lattices. This problem can be reduced to an integral equation, where the effect of the interaction of the neighboring inclusions is taken into account by replacing the free space kernel (electric potential due to a unity charge), by an infinite sum of electric potentials. The main contribution of our study is to recognize, that if we add to the previous kernel another term, we will obtain a periodic function. This approach has the following advantages: first, the computational effort to solve the integral equation is greatly reduced, and furthermore the effect of arbitrarily high distance inclusions can be taken into account without appreciable increase in the computation time. This is not true for the standard approach, where the computation time grows with (2N)/sup 3,/ N being the side of the cube containing the indices of the inclusions taken into account. Secondly, with our formulation it is possible to calculate unambiguously the interaction constants for the periodic lattice.