  Modal expansions are useful to understand wave propagation in an infinite
electromagnetic transmission line or waveguide. They can also be used to
construct generalized Dirichlet-to-Neumann maps that can be used to provide
artificial boundary conditions for truncating a computational domain when
discretizing the field by finite elements. The modes of a waveguide arise as
eigenfunctions of a non-symmetric eigenvalue problem, and the eigenvalues
determine the propagation (or decay) of the modes along the waveguide. For the
successful use of waveguide modes, it is necessary to know that the modes exist
and form a dense set in a suitable function space containing the trace of the
electric field in the waveguide. This paper is devoted to proving such a
density result using the methods of Keldysh. We also show that the modes
satisfy a useful orthogonality property, and show how the Dirichlet-to-Neumann
map can be calculated. Our existence and density results are proved under
realistic regularity assumptions on the cross section of the waveguide, and the
electromagnetic properties of the materials in the waveguide, so generalizing
existing results.
