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. Sec. Am. A 13, 1870 (1996)] has shown that the truncated Fourier series of products o f discontinuous functions that were used in the differential theory of grat ings during the past 30 years are not converging everywhere in TM polarizat ion. They turn out to be converging everywhere only at the limit of infinit ely 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 h armonics used to represent the field. The fast convergence is observed on b oth metallic and dielectric gratings with sinusoidal, triangular, and lamel lar profiles as well as with cylindrical and rectangular rods, and examples are shown on gratings with 100% modulation. The new formulation opens a ne w wide range of applications of the method, concerning not only gratings us ed in TM polarization but also conical diffraction, crossed gratings, three -dimensional problems, nonperiodic objects, rough surfaces, photonic band g aps, nonlinear optics, etc. The formulation also concerns the TE polarizati on case for a grating ruled on a magnetic material as well as gratings rule d on anisotropic materials. The method developed is applicable to any theor y that requires the Fourier analysis of continuous products of discontinuou s periodic functions; we propose to call it the fast Fourier factorization method. (C) 2000 Optical Society of America [S0740-3232(00)01910-4] OCIS co des: 050.0050, 050.1940, 050.1970, 050.2770, 260.2110.