Convergence analysis of various factorization rules in the Fourier-Bessel basis for solving Maxwell equations using...

Area of Science:

• Computational Electromagnetics
• Numerical Analysis
• Optics and Photonics

Background:

• Convergence of solutions to Maxwell equations depends on factorization rules, especially in orthogonal expansions.
• While Fourier basis (plane-wave expansion) is well-studied, other bases like Fourier-Bessel (cylindrical-wave expansion) need more attention.
• Existing research on Fourier-Bessel basis suggests inverse factorization rules improve convergence, but overlook other possibilities.

Purpose of the Study:

• To mathematically explore and compare four distinct factorization rules for Maxwell equations in cylindrical coordinates.
• To analyze the convergence of these rules using the Fourier-Bessel expansion in both infinite and finite domains.
• To identify the optimal factorization rule for the fastest convergence in the modal method with Fourier-Bessel basis.

Main Methods:

• Mathematical derivation and demonstration of four different factorization rules.
• Application of Fourier-Bessel expansion in cylindrical coordinates for solving Maxwell equations.
• Convergence comparison using a step-index fiber with a known exact solution and various VCSEL structures.

Main Results:

• Demonstrated four factorization rules for Fourier-Bessel expansion in cylindrical coordinates.
• Showed that cylindrical-wave expansion differs significantly from plane-wave expansion.
• Identified that inverse factorization rules can degrade numerical convergence for electric fields perpendicular to discontinuities.

Conclusions:

• The optimal factorization rule depends on the specific basis and coordinate system used.
• An inverse factorization rule is not universally superior and can harm convergence in certain scenarios.
• This study identifies the most effective factorization rule for achieving rapid convergence with the Fourier-Bessel basis in optical simulations.