Implementation of the Perfectly Matched Layer to Determine the Quality Factor of Axisymmetric Resonators in COMSOL
Abstract: Due to the inseparability of the wave equation, numerical methods are needed to develop an accurate electromagnetic model for various axisymmetric optical resonators such as micro-discs and micro-toroids. A COMSOL model for axisymmetric resonators has already been developed, however that model lacks the capability to determine the quality factor of a micro-cavity precisely. Here our purpose is the implementation of a perfectly matched layer to determine the quality factor of axisymmetric resonators with high accuracy in COMSOL.
Perfectly matched layers (PML) act as artificial boundaries to truncate the computation domain of open region scattering problems in the _nite element method. The whispering gallery modes (WGM) of an open optical micro-cavity radiate into surroundings and a PML is required in order to block the unwanted reections from the boundaries of the computation domain. One research group  has developed a COMSOL model for open axisymmetric resonators without applying any transverse approximation to the wave equation. However in their model no PML has been implemented and as a result the WGM quality factor can not be determined accurately. In that model, the quality factor due to the WGM radiation has been estimated by placing a bound on its minimum and maximum possible values. Determination of the quality factor with high accuracy is important in certain applications such as cavity ring down spectroscopy where decay time depends upon the quality factor. For accurate determination of the WGM quality factor we have implemented PML along the boundaries of the computation domain. We treat the PML as an anisotropic absorber and implement it in the cylindrical coordinate system. Our model is applicable to any axisymmetric resonator geometry but due to the availability of analytical expressions for spherical resonators, we have tested our model by determining the quality factors of a silica micro-sphere in air. We have found that our simulation results are in excellent agreement with the analytical results.