It is not essential to produce Both the E and H fields.  Either one can
produce the other.  For example the toroidal antenna launches only an E
field with an associated vector potential that produce, in the far field,
an H field to accompany the E field.  Similarly a loop antenna launches
only an H field with its associated vector potential and as a result the E
field forms in the far field.  In both cases one ends up with a complete
propagating electromagnetic field.