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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 thesis. Its solution is given by the scattering-matrix (S-matrix) to be calculated. The fulfillment of the boundary conditions is possible in different ways: The Conventional Differential Method is based on the Shooting Method. Its disadvantage is numerical integration along exponentially increasing functions (anti-evanescent waves). As naturally stable pendant to this, the Direct S-matrix Integration has been developed in the context of this thesis: It integrates the S-matrix directly by a differential equation derived for this. Both methods have O(N³)-complexity, with N the number of harmonics. Recently, alternative rigorous Fourier methods of merely O(N·ln(N))-complexity based on integral equations are emerging. Improvable points here are the number of iterations for the solution of the integral equation, the memory consumption and a lack of flexibility. One remedy to this developed in the context of this thesis is the S-vector algorithm. The presented methods are combined with polarization ray tracing as well as the angular spectrum of plane waves outside gratings. The specified examples show the need, applicability and merit of rigorous Fourier methods.

In the present we investigate the advantages of the phase lag analysis for the derivation of phase-fitted techniques on several numerical schemes. Relying on the main characteristics of the phase lag we evaluate the parameters needed firstly for Runge-Kutta methods and secondly for high order variational integration methods, so that the phase lag and its derivatives are zero. The proposed methods are tested for the solution of initial value problems on ordinary differential equations of second order, like the Henon-Heiles model.