Direct integration of the S-matrix applied to rigorous diffraction

A novel Fourier method for rigorous diffraction computation at periodic structures is presented. The procedure is based on a differential equation for the S-matrix, which allows direct integration of the S-matrix blocks. This results in a new method in Fourier space, which can be considered as a numerically stable and well-parallelizable alternative to the conventional differential method based on T-matrix integration and subsequent conversions from the T-matrices to S-matrix blocks. Integration of the novel differential equation in implicit manner is expounded. The applicability of the new method is shown on the basis of 1D periodic structures. It is clear however, that the new technique can also be applied to arbitrary 2D periodic or periodized structures. The complexity of the new method is O (N-3) similar to the conventional differential method with N being the number of diffraction orders.