Fully stable numerical calculations for finite one- dimensional 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. PACS: 78.20.Bh; 78.20.Ci; 42.25.Bs OCIS codes: (000.3860) Mathematical methods in physics; (000.4430) Numerical approximation and analysis; (260.2110) Electromagnetic optics; (350.4238) Nanophotonics and photonic crystals.