Numerical instability of the C method when applied to coated gratings and methods to avoid it

Abstract

We recently found that the coordinate transformation method (the C method) equipped with well-established recursive algorithms for solving the system of linear equations is numerically instable when it is applied to thinly coated gratings. The origin of this new kind of numerical instability is not the exponential dependence of the field in the coated layers but the ill condition of the eigenvector matrix of the C method when the truncation number is high. Two simple and effective methods to circumvent the new instability are recommended. We also found that the popular recursive matrix algorithms have different (poor) immunities to the new instability, and they all perform inferiorly to the full matrix (nonrecursive) algorithm.

Dynamical metasurfaces: electromagnetic properties and instabilities

“…The differential formalism can handle both shallow and deep structures; however, we need to adopt local coordinate distortion instead of the global coordinate translation as studied in a series of studies [182][183][184][185][186][187][188][189][190][191]. That is because the local transformation affects only a bounded region near the structured surface and enables us to apply a simple criterion to select the incoming and outgoing fields.…”

“…gA > 1), large matrix elements do not localise near the diagonal elements any longer. This phenomenon occurs even in the conventional grating calculation handled by the differential formalism [182][183][184][185][186][187][188][189][190][191][192][193] as mentioned at the beginning of this chapter. In that regime, we have to adopt local coordinate distortion instead of the global coordinate translation (7.1).…”

Čerenkov radiation in vacuum from a superluminal grating

“…Our previous work is based on the coordinate translation method originally proposed by Chandezon et al [30,31], where they consider static corrugated interfaces and match Maxwell's boundary conditions directly at the interfaces with the help of differential geometry. Using this method, it is straightforward to take the structure of the interfaces into consideration, and it has been utilised to calculate structured surfaces of various media, including anisotropic, plasmonic and dielectric materials [32][33][34][35][36] It is also worth noting that there is a series of studies, which confirm that the method works well for smooth shallow corrugations and propose possible ways to improve the method so that they can handle deep corrugation even with sharp edges [37][38][39][40][41][42][43]. In these works, local distortion of the coordinate systems is applied instead of the global translation of the coordinate in order to improve the convergence.…”