Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings

Abstract

Two recursive and numerically stable matrix algorithms for modeling layered diffraction gratings, the S-matrix algorithm and the R-matrix algorithm, are systematically presented in a form that is independent of the underlying grating models, geometries, and mountings. Many implementation variants of the algorithms are also presented. Their physical interpretations are given, and their numerical stabilities and efficiencies are discussed in detail. The single most important criterion for achieving unconditional numerical stability with both algorithms is to avoid the exponentially growing functions in every step of the matrix recursion. From the viewpoint of numerical efficiency, the S-matrix algorithm is generally preferred to the R-matrix algorithm, but exceptional cases are noted.

Differential theory of diffraction in cylindrical coordinates

“…The differential method of light diffraction that has found numerous applications in grating studies [1][2][3][4][5][6][7][9][10][11][12][13][14][15], waveguide optics, Raman scattering [33], is successfully applied in cylindrical geometry to the study of diffraction by circular apertures in perfectly or finitely conducting screens, which permitted determining the role of the surface plasmon in the near-field distribution and the importance of the finite conductivity in increasing the directivity of the far-field radiation pattern. The method can successfully compete with other known methods working in Cartesian coordinates, like the adaptive-mesh FDTD [34] or the Fourier-series method with perfectmatching layers (PML) [35].…”

“…This can be avoided by suitable renormalization in the case of plane wave expansion, but this is impossible when the explicit form of the field is unknown, as it happens in case of inhomogeneous media. During the '90s, this problem was definitely solved in electromagnetism by applying [15] a propagation algorithm (called S-matrix algorithm) already known in other domains of science [16,17]. It is easy to understand the idea by slightly changing eq.…”

“…To solve this difficulty, it is necessary to divide the inhomogeneous domain into subdomains (slices), small enough so that it is possible to determine the T (j) -matrix for each slice (numbered j) without numerical instability. A simple recurrence formula [15] enables then to determine the S (j) -matrix for each set of slices starting from the first one (j = 1) of the set: …”

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

“…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 is applicable for infinitely periodic structures and originated in the diffractive optics community more than 30 years ago (Knop, 1978). In the past decades the method has matured due to fundamental studies and improvements to its stability (Moharam et al, 1995b;Li, 1996a) and convergence (Granet and Guizal, 1996;Lalanne and Morris, 1996;Li, 1996b). Other important contributions to the evolution of the method are the techniques of adaptive spatial resolution (Granet, 1999) and normal vector fields Nevière, 2000, 2001;Schuster et al, 2007).…”