Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation

Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method

“…The coupled linear systems (47) for l = 1, ..., M − 1 have to be solved in a stable manner by using a non-homogeneous S-matrix algorithm (Pisarenco et al, 2011) which is an adaptation of the classical (homogeneous) S-matrix algorithm to the contrast field formulation.…”

“…The interface relation (47a) is unstable in the sense that its direct use for sequential elimination of field amplitudes leads to large round-off errors (Ko and Inkson, 1988;Li, 1996a;Pisarenco et al, 2011). To avoid instability, we use an S-matrix representation.…”

“…It can be determined from the T-matrix T l,l+1 and the vector g ′ l,l+1 (see (Pisarenco et al, 2011)),…”

“…We would like to find the pair (S a,c , f a,c ) which realizes the mapping for layers a and c It is shown in (Pisarenco et al, 2011) that (S a,c , f a,c ) is given by…”

“…The AFMM-CFF (Pisarenco et al, 2010(Pisarenco et al, , 2011) is a recent extension of the FMM to aperiodic structures. It builds upon the aperiodic Fourier modal method developed and refined by Lalanne and co-workers (Lalanne and Silberstein, 2000;Silberstein et al, 2001;Cao et al, 2002;Hugonin and Lalanne, 2005;Lecamp et al, 2007) with the key difference that the AFMM-CFF solves Maxwell's equations formulated in terms of a contrast (scattered) field instead of a total field.…”