Convergence of Factorization Rules for Modal Methods in Fourier-Bessel Basis. Is the Inverse Rule Really the Best?

The choice of factorization rule can strongly affect the convergence of solutions to Maxwell equations based on the orthogonal expansion of electromagnetic fields. While this issue has already been investigated thoughtfully for the Fourier basis (plane-wave expansion), for other bases it has not yet received much attention. Although there are works showing that, in the case of the Fourier-Bessel basis (cylindrical wave expansion), use of an inverse factorization rule can provide faster convergence than Laurent’s rule, these works neglect the fact that other rules are also possible. In this work, I demonstrate some different factorization rules for solving Maxwell equations in cylindrical coordinates using Fourier-Bessel expansion and I compare their convergence for a step-index fiber (which has a known exact solution and thus enables the absolute numerical error to be determined), as well as for several VCSEL structures. This allows to identify the factorization rule that gives the fastest convergence of the modal method using the Fourier-Bessel basis.