Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence

A new formulation of the differential method in TM polarization, based on correct representation of truncated Fourier series of products of discontinuous functions, is proposed. Although the derived equations are more complicated than in the classical formulation, the convergence rate with respect to the truncation parameter (with the number of diffraction orders taken into account) is much faster for arbitrary grating profiles, approaching the convergence rate in TE polarization. Numerical examples are presented for dielectric and metallic sinusoidal gratings with a 100% modulation ratio.