The resolution of a dipole field into transverse electric and transverse magnetic waves

The problem is considered of representing the vacuum electromagnetic field of an arbitrarily orientated dipole as the superposition of coplanar transverse magnetic (TM) and transverse electric (TE) fields. The method is to express the dipole field as an angular spectrum of plane waves, and to introduce the appropriate representation of each plane wave of the spectrum. For a time-harmonic electric dipole, in the non-trivial case in which the components of the magnetic and electric vectors, respectively, of the TM- and TE-fields vanish in a direction perpendicular to the dipole, the TM- and TE-fields are given explicitly in terms of elementary functions.