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

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.