Numerical stability of the C method and a perturbative preconditioning technique to improve convergence

Abstract

The translational coordinate transformation method (the C method) in grating theory is studied numerically and analytically. We first study the convergence characteristics of the C method by numerical computations in high floating-point data precisions. Guided by insights gained from this numerical study we analytically studied condition numbers of the most important eigenvalues of the eigenvalue problem of the C method. Asymptotic estimates of condition numbers of these eigenvalues and estimates of convergence rate of the error in satisfying the Helmholtz equation by the eigenvectors are derived. These theoretical results explain well many observed numerical phenomena of the C method. Using the first-order perturbation theory of simple eigenvalues we analyze the effects of round-off errors on eigenvalue distribution and condition numbers. This leads to an extremely simple perturbative preconditioning technique that significantly improves the numerical stability of the C method with as little as just one line of code modification. The performance of the perturbatively preconditioned C method is not inferior to the C method preconditioned by the multilinear parameterization technique. We recommend it as the preferred method for modeling deep and smooth gratings.

Diffraction by gratings: from the C-method to the stochastic C-method

“…The high sensitivity of diffraction to geometry can be highlighted by comparing the results obtained with the C method for two gratings with very similar shapes: P1 with profile a 1 (x) = 1 − sin 6 πx/2 4 and P2 with profile a 2 (x) = 1/ 1 + (2x) 8 .…”

“…This wall manifests as poor matrix conditioning when writing boundary conditions at the surface of the grating, rendering inversions impossible. Various methods have been developed to overcome these challenges, such as analytical regularization [4], or increasing the calculation precision, using a parametric coordinate system [5], or working with multi-precision [6] [7] [8].…”

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.…”