Expansion of an arbitrary field in terms of waveguide modes

The problem of expanding a field over the set of waveguide modes is well known. Nevertheless, one may find small differences in the way this concept is used. Some employ a superposition of waveguide modes that include all field components, and a mode is considered as a single entity which propagates undisturbed along the structure. Others prefer to expand only the transverse-field components, whereas the longitudinal ones are derived from Maxwell's equations. It is shown that the latter is correct, at least for the important class of 2-dimensional structures. The two approaches coincide, however, if the structure is the waveguide for which the set of modes was calculated. The formal mathematical proof is restricted to a nonlossy medium. It is shown that the modes with real propagation coefficients squared suffice to construct a complete set. Depending on the boundary conditions in the longitudinal z direction, some or many of these modes may or may not be needed.