Grating theory: new equations in Fourier space leading to fast converging results for TM polarization

Using theorems of Fourier factorization, a recent paper [J. Opt. Soc. Am. A 13, 1870 (1996)] has shown that the truncated Fourier series of products of discontinuous functions that were used in the differential theory of gratings during the past 30 years are not converging everywhere in TM polarization. They turn out to be converging everywhere only at the limit of infinitely low modulated gratings. We derive new truncated equations and implement them numerically. The computed efficiencies turn out to converge about as fast as in the TE-polarization case with respect to the number of Fourier harmonics used to represent the field. The fast convergence is observed on both metallic and dielectric gratings with sinusoidal, triangular, and lamellar profiles as well as with cylindrical and rectangular rods, and examples are shown on gratings with 100% modulation. The new formulation opens a new wide range of applications of the method, concerning not only gratings used in TM polarization but also conical diffraction, crossed gratings, three-dimensional problems, nonperiodic objects, rough surfaces, photonic band gaps, nonlinear optics, etc. The formulation also concerns the TE polarization case for a grating ruled on a magnetic material as well as gratings ruled on anisotropic materials. The method developed is applicable to any theory that requires the Fourier analysis of continuous products of discontinuous periodic functions; we propose to call it the fast Fourier factorization method.