Rigorous Full Vectorial Analysis of Electromagnetic Wave Propagation in 1d

We propose a new approach to solve the problem of the propagation of electromagnetic waves in unidimensional media with an arbitrary variation of their dielectric permittivity. This method is deduced from the Maxwell equations with a minimum of approximations and allows a full vectorial description of both the electric and magnetic fields through the direct calculation of their Cartesian coordinates. The problem is then equivalent to the solution of a pair of uncoupled ordinary differential equations. We use a very intuitive, highly accurate, pseudospectral technique to solve these equations. This pseudospectral method is based in a combination of Fourier and polynomial expansions of the solution providing very good precision and excellent stability with a relatively low computational effort. We present a simple model of a photonic crystal as an example of application of this technique to real electromagnetic problems.