ABSTRACT
Maxwell equations are applied to periodic media, they can be projected onto a Fourier basis with respect to one or more coordinates. This procedure is quite useful, because it reduces the order of the differential equations. In the case of diffraction gratings used in in-plane mounts, the Fourier decomposition enables to transform Maxwell equations into a set of ordinary differential equations, which can be integrated using well-developed numerical methods. When using a truncated Fourier series, the truncation can cause numerical problems for highly conducting materials in TM polarization. Another numerical problem appears in both TE and TM cases, due to the exponentially growing terms during the integration. Modern science has pointed out several domains in which an extremely small error in an initial value of a solution of an equation produces unpredictable results. Well-known examples are meteorology, or the theory of deterministic chaos.
