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 andvarpi_{0}are constants andF(theta)is any function oftheta. 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 abouttheta=0. It appears that the pattern converges to the characteristic pattern as the frequency is raised, if a is notinfty, and that the impedance converges to the characteristic impedance for alla.