One-dimensional finite-elements method for the analysis of whispering gallery microresonators.

Abstract

By taking advantage of axial symmetry of the planar whispering gallery microresonators, the three-dimensional (3D) problem of the resonator is reduced to a two-dimensional (2D) one; thus, only the cross section of the resonator needs to be analyzed. Then, the proposed formulation, which works based on a combination of the finite-elements method (FEM) and Fourier expansion of the fields, can be applied to the 2D problem. First, the axial field variation is expressed in terms of a Fourier series. Then, a FEM method is applied to the radial field variation. This formulation yields an eigenvalue problem with sparse matrices and can be solved using a well-known numerical technique. This method takes into account both the radiation loss and the dielectric loss; hence, it works efficiently either for high number or low number modes. Efficiency of the method was investigated by comparison of the results with those of commercial software.