Accurately calculating high Q factor of whispering-gallery modes with boundary element method

A new method based on the well-studied boundary element method (BEM) is introduced to calculate ultra-high quality factors (Q) of whispering-gallery modes (WGMs) by studying the Poyting vector. In traditional numerical method, Q can be obtained through the formula Q = -Re(k)/2Im(k) by solving the wavenumber (k) of resonances, such that the precision of Q is determined by the wavenumber k. On the other hand, the precision of the numerical simulations is limited by the computation ability of computer, and the reported simulations of high Q modes are about 10⁵-10⁶ based on the popular PC. In this paper, since the field distribution of WGMs has much higher precision compared to the wavenumber k, we can alternatively evaluate the Q by solving the energy in microcavity and the energy through radiation lost (Poyting vector), in the other words, Q = 2pi*(stored energy) / (energy lost per cycle). As a result, the new method has a very high calculation accuracy, which exceeds the limitation in Q calculation of traditional methods. As an example, with this method, a circular microcavity, which has strict analytical solution for WGMs, is investigated. We demonstrate that the present method can evaluate Q factor up to 10¹⁰, while a little computation resource is required.