Isotropic Radiators

Can the radiation pattern of an antenna be isotropic? A simple argument suggests that this is difficult. The intensity of radiation depends on the square of the electric (or magnetic) field. To have isotropic radiation, it would seem that the magnitude of the electric field would have to be uniform over any sphere in the far zone. However, the electromagnetic fields in the far zone of an antenna are transverse, and it is well known that a vector field of constant magnitude cannot be everywhere tangent to the surface of a sphere (Brouwer’s “hairy-ball theorem” [1]). Hence, it would appear that the transverse electric field in the far zone cannot have the same magnitude in all directions, and that the radiation pattern cannot be isotropic, IF the radiation is everywhere linearly polarized [2, 3]. However, electromagnetic waves can have two independent states of polarization, described as elliptical polarization in the general case. While a transverse electric field with a single, linearly polarized component cannot be uniform over a sphere in the far zone, it may be possible that the sum of the squares of the electric fields with two polarizations is uniform.