  Polygon and star modes enable unidirectional emission and single-frequency
lasing in whispering gallery microcavities. To understand their properties and
facilitate design, we have adopted both two-dimensional and three-dimensional
full-wave perturbation methods to simulate these modes. Our simulation
demonstrates that a tapered optical fiber can be used as a weak perturbation to
coherently combine multiple whispering gallery modes into a polygon or star
mode. Additionally, our simulation predicts an optical quality factor as high
as $10^7$ for the polygon modes, which is in good agreement with the experiment
results.
