# Stable and Efficient Rigorous Fourier Methods

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Rigorous Fourier methods are methods for the rigorous calculation of the scattering of waves at gratings which are based on Fourier expansions of the field and material distribution in the direction(s) of periodicity; they are of importance for the calculation of optical systems containing diffractive elements – e.g. interferometers. In the direction(s) of periodicity, the time-independent Maxwell equations are projected onto a Fourier basis. The remaining ordinary differential equation defines together with the boundary conditions at the homogeneous medium of incidence and transmission the boundary value problem, the subject of this work. Its solution is given by the scattering-matrix (S-matrix) to be calculated – or the scattering vector (S-vector) respectively for only one given incidence scenario. This work resumes already known, scattering computation techniques such as the S-matrix algorithm as well as integral equation based methods and subsequently develops the direct S-matrix integration as well as the S-vector algorithm as novel, alternative concepts. The specified examples show the need and merit of efficient rigorous Fourier methods.