Fully stable numerical calculations for finite onedimensional structures: mapping the Transfer Matrix method

We design a fully stable numerical solution of the Maxwell´s equations with the Transfer Matrix Method (TMM) to understand the interaction between an electromagnetic field and a finite, one-dimensional, nonperiodic structure. Such an exact solution can be tailored from a conventional solution by choosing an adequate transformation between its reference systems, which induces a mapping between its associated TMMs. The paper demonstrates theoretically the numerical stability of the TMM for the exact solution within the framework of Maxwell´s equations, but the same formalism can efficiently be applied to resolve other classical or quantum linear wave-propagation interaction in one, two, and three dimensions. This is because the formalism is exclusively built up for an in depth analysis of the TMM´s symmetries