Justification of matrix truncation in the modal methods of diffraction gratings

In the Fourier modal method (FMM) and Chandezon method for modelling diffraction gratings, the reduction method is used to obtain approximate numerical solutions of certain matrix eigenvalue problems in the infinite Fourier space. However, at least in the optics literature to date, the legitimacy of using the reduction method in these situations has not been carefully studied. By using the classical theory of determinants of infinite order, I show that the solutions of the eigenvalue problems with truncated matrices tend to the exact solutions as the matrix order increases; thus the use of the reduction method is legitimate. This conclusion is valid for any permittivity variations (continuous or discontinuous) in the case of the FMM, and for any permissible grating profiles (smooth or edgy) in the case of the Chandezon method.