Spectral reconstruction in Fourier modal methods: on exclusive robustness of Gegenbauer method

Fourier spectral methods constitute a rigorous memory-minimizing technology for the analysis of photonic structures. However, the discrete geometry of these artifacts can lead to the Gibbs phenomenon, and consequently, a dramatic deterioration of the exponential rate convergence. In this paper, we study the applicability of spectral reconstructions for the resolution of the Gibbs phenomenon in Fourier modal methods. We show in practice, the emergence and propagation of a computational disorder potentially incapacitate most of the popular post-processing techniques from retrieving a Fourier spectral solution in the vicinity of discontinuities. Nonetheless, we numerically substantiate and rigorously prove that the classical Gegenbauer method remains exceptionally robust against this phenomenon. Specifically, we show the Gegenbauer reconstruction error at worst diminishes, proportional to O(N^{-1}) for the Fourier order N, while the convergence can be even exponentially fast up to a constant summand.