E. Popov et M. Neviere, Grating theory: new equations in Fourier space leading to fast converging results for TM polarization, J OPT SOC A, 17(10), 2000, pp. 1773-1784
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.