Ideal efficiency of a radiating element in an infinite array

The efficiency of a radiating element in an infinite planar array has been defined as the ratio of the power radiated to the power available from the element. This efficiency is a measure of the strength of mutual coupling effects and also of the average reflection coefficient magnitude at a typical element in the active array as the elements are phased over a prescribed range. An upper bound to this efficiency, which is only a function of the array geometry, exists and corresponds to an array of antennas with certain ideal characteristics. A technique for evaluating this ideal or maximum efficiency of an element in an arbitrary infinite lattice is presented. While this efficiency is readily evaluated for some special configurations, the computation for any general space lattice becomes straightforward in terms of the formalism. The formalism is based on the "reciprocal lattice" and a transformed (Cartesian) version of the space and reciprocal lattices. The required integral is numerically equal to the area common to a unit square and an ellipse in this transformed lattice. Results are given for various rhombic and parallelogram lattices which include the square and the equilateral triangle as special cases. It is noted that for parallelograms of aspect ratio greater than about 4 to 1, the efficiency is nearly the same as for the limiting linear array.