The far-field operator for an exterior boundary value problem for the Helmholtz equation maps the boundary data onto the far-field pattern of the solution. This paper computes the L2-adjoint of this operator for various choices of boundary conditions. In scattering theory the boundary data are given by the traces of plane waves. We characterize the closure of the span of the images of these plane waves under the far field operator.
Finally, the results are extended to more general topologies including Sobolev and Hölder norms.