Degrees of freedom and maximum directivity of antennas: a bound on maximum directivity of nonsuperreactive antennas

We derive a general fundamental directivity limitation formula that applies to nonsuperreactive antennas of any size that fit within a minimum sphere of any given radius rmin. The derivation is done by using a new concept: the degrees of freedom (DoF) of the field radiated by arbitrary sources within the minimum sphere must be twice the maximum directivity. The formula converges to the known bound of the directivity for large rmin. For small spheres, it becomes equal to three, i.e., 4.8 dBi, which is the directivity of the Huygens source. The transition region between these two limiting cases is determined by counting the most significant spherical modes at the surface of the minimum sphere. This is not trivial, because spherical modes have a gradual cutoff when their order approaches krmin. Therefore, we use a weighted summation where the weighting factor is inversely proportional to the radiation-Q of the modes. This extends the DoF from a discrete to continuous function of the minimum sphere radius. The final maximum directivity is similar to a previously published heuristic limit. The formulas are compared to results for measured antennas with large directivity and superdirectivity.