Dense Sets and Far Field Patterns in Electromagnetic Wave Propagation

It is shown that the electric far field patterns corresponding to the scattering of entire incident fields by a bounded perfectly conducting obstacle are dense in the space of square integrable tangential vector fields defined on the boundary of the unit ball if and only if there does not exist a Maxwell eigenfunction that is an electromagnetic Herglotz pair, i.e. a solution \ E,H\ of Maxwell’s equations defined in all of space such that \[ {lim }r → ∞ 1/r|x| r {( {|E(x)|^2 + |H(x)|^2 } )dx < ∞ }.\]