Plain models of very simple waveguide junctions without any solution for very rich sets of excitations Paolo Fernandes * Mirco Raffetto † January 26, 2009 Abstract 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. Keywords: electromagnetic theory; guided waves and wave-guiding structures; metamaterials; electromagnetic boundary value problems; ill posedness; electromagnetic simulators. 1 Introduction In any branch of the science we deal invariably with simplified models of the “real world”, rather than with the “real world” itself (whatever this more philosophical than scientific term may mean!). After all, even physical laws are nothing but particularly well established models accepted by the whole scientific community. A model always catches the features of the phenomena under consideration that are essential in the context where the model itself is to be used, while neglecting finer details. This approach is both unavoidable and fruitful and most of the progress of the science has been based on it. Familiar examples of that in electromagnetics are many models of radiation phenomena and microwave circuits usually considered in applications, numerical simulations and education [1], [2]. In many scientific communities, good properties of models are often taken for granted. In particular, most of them are implicitly supposed to be “well posed problems”, that is to say, to have a unique solution, which continuously depends on the data [3], [4]. It is sometimes argued that, since a physical system driven by a specific excitation will behave in a specific way, its model should automatically have one and only one solution for any allowed excitation. This is not a good argument, however. Since any model neglects something, it cannot be taken for granted that it behaves like the corresponding physical system, not even approximately. As a matter of fact, as we will see later on, very reasonable simplifying assumptions may lead, on occasion, to * Istituto di Matematica Applicata e Tecnologie Informatiche del Consiglio Nazionale delle Ricerche, Via De Marini 6, I–16149 Genoa, Italy, email: fernandes@ge.imati.cnr.it † Department of Biophysical and Electronic Engineering, University of Genoa, Via Opera Pia 11a, I–16145, Genoa, Italy, email: raffetto@dibe.unige.it 1