Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings.

We mathematically prove and numerically demonstrate that the source of the convergence problem of the analytical modal method and the Fourier modal method for modeling some lossless metal-dielectric lamellar gratings in TM polarization recently reported by Gundu and Mafi [J. Opt. Soc. Am. A 27, 1694 (2010)] is the existence of irregular field singularities at the edges of the grating grooves. We show that Fourier series are incapable of representing the transverse electric field components in the vicinity of an edge of irregular field singularity; therefore, any method, not necessarily of modal type, using Fourier series in this way is doomed to fail. A set of precise and simple criteria is given with which, given a lamellar grating, one can predict whether the conventional implementation of a modal method of any kind will converge without running a convergence test.