Abstract

This paper presents a new formulation for the fast and accurate three-dimensional vectorial analysis of dielectric whispering-gallery-mode (WGM) resonators using the finite element method (FEM). This method relies on two properties of the WGM resonators: (1) the axial symmetry to reduce the number of coordinate variables from three to two and (2) the use of the divergence equation for the magnetic field and its axial symmetry to express the azimuthal magnetic component of the field in terms of its transverse components. As a result, for our FEM analysis, only the transverse part of the magnetic Helmholtz equation is required, which is composed of two field variables and two coordinate variables. The method can use scalar elements without generating any spurious modes and without the need for a penalty function. This can substantially reduce the computational time and complexity. After verifying this FEM formulation through comparing the simulation results with the known analytical solutions for a few special resonators, a variety of WGM resonators in the silicon photonic platform are analyzed in terms of key parameters needed for designing integrated photonic structures.

© 2017 Optical Society of America

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