Modelling sub-wavelength ultra-high-Q optical resonators using time domain numerical methods

In this paper, we discuss challenges in modelling circular geometries in Cartesian time domain numerical methods. The main focus of this paper is to reduce the apparent shift in the deduced resonance frequencies. We show that this shift is completely dependent on the discretisation technique and hence propose a resolution to minimise it. In addition, we discuss the limitations of extracting resonances from the time domain signal obtained from such a numerical model using a FFT and compare complex frequency extraction techniques. We show that, despite its high computational complexity, a modified difference Prony method provides best accuracy. We use this method to show that our proposed method of discretising to maintain resonator area is uniformly convergent with mesh refinement.