Resolving the Gibbs phenomenon via a discontinuous basis in a mode solver for open optical systems

TL;DR

This paper introduces a hybrid basis combining smooth and discontinuous functions to effectively resolve the Gibbs phenomenon in mode solvers for open optical systems, significantly improving convergence and eliminating spurious solutions.

Contribution

The authors develop a novel mode solver that incorporates a discontinuous basis to accurately and efficiently handle discontinuities in open electromagnetic systems.

Findings

Exponential convergence of discontinuous basis in resolving Gibbs phenomenon

Elimination of spurious solutions in mode calculations

Enhanced simulation speed and accuracy for open optical systems

Abstract

Partial differential equations are frequently solved using a global basis, such as the Fourier series, due to excellent convergence. However, convergence becomes impaired when discontinuities are present due to the Gibbs phenomenon, negatively impacting simulation speed and possibly generating spurious solutions. We resolve this by supplementing the smooth global basis with an inherently discontinuous basis, incorporating knowledge of the location of the discontinuities. The solution's discontinuities are reproduced with exponential convergence, expediting simulations. The highly constrained discontinuous basis also eliminates the freedom to generate spurious solutions. We employ the combined smooth and discontinuous bases to construct a solver for the modes of a resonator in an open electromagnetic system. These modes can then expand any scattering problem for any source configuration or incidence condition without further numerics, enabling ready access and physical insight into the spatial variation of Green's tensor. Solving for the modes is the most numerically intensive and difficult step of modal expansion methods, so our mode solver overcomes the last major impediment to the use of modal expansion for open systems.