Frequency independent antennas

There is a class of antennas whose pattern as well as impedance is practically independent of frequency for all frequencies above a certain value. The general formula for their shape isr = e^{a}(\varpi+\varpi_{0})F(\theta)wherer \theta \varpiare the usual spherical coordinates, a and\varpi_{0}are constants andF(\theta)is any function of\theta. Assuming a to be positive,\varpiranges from -\inftyto some finite value which determines the low frequency limit. For such antennas a change of frequency is equivalent to a rotation of the antenna about\theta=0. It appears that the pattern converges to the characteristic pattern as the frequency is raised, if a is not\infty, and that the impedance converges to the characteristic impedance for alla.