Plain Models of Very Simple Waveguide Junctions Without Any Solution for Very Rich Sets of Excitations

Almost trivial waveguide junctions involving standard media and metamaterials modelled by effective constitutive parameters are investigated. It is shown that, when some dielectric configurations are present, no solution can be found for these models, for some excitations on the ports which are very regular, are not at all pathological and allow simple modal expansions. The set of excitations for which no solution exists is very rich and contains excitations almost indistinguishable from those for which the solution exists. This lack of solution does not originate, as usual in electromagnetics, from the excitation of a resonance of an ideal cavity. It rather arises from a mechanism similar to the one that causes ill posedness of inverse problems. What is new and unexpected is to find this kind of ill posedness in a direct problem. The well known modal technique is exploited heavily but, quite unusually, to prove the non-existence of solutions rather than to find them. Finally, the importance of results on the
a priori
well posedness of models is pointed out.