Perturbative analytic theory of an ultrahigh-Qtoroidal microcavity

Bumki Min, Lan Yang, Kerry Vahala
2007 Physical Review A. Atomic, Molecular, and Optical Physics  
A perturbation theoretic approach is proposed as an efficient characterization tool for a tapered fiber coupled ultrahigh-quality factor ͑Q͒ toroidal microcavity with a small inverse aspect ratio. The Helmholtz equation with an assumption of quasi-TE/TM modes in local toroidal coordinates is solved via a power series expansion in terms of the inverse aspect ratio and the expanded eigenmode solutions are further manipulated iteratively to generate various characteristic metrics of the
more » ... toroidal microcavity coupled to a tapered fiber waveguide. Resonance wavelengths, free spectral ranges, cavity mode volumes, phase-matching conditions, and radiative Q factors are derived along with a mode characterization given by a characteristic equation. Calculated results are in excellent agreement with full vectorial finite-element simulations. The results are useful as a shortcut to avoid full numerical simulation, and also render intuitive insight into the modal properties of toroidal microcavities. A local toroidal coordinate system ͑r , , ͒ that is intuitively suitable for electromagnetic problems with a toroidal geometry will be used throughout the rest of this paper. Figure 1 shows the local toroidal coordinate system with a rendering of the toroidal microcavity structure ͑D =2R: major diameter, d =2a: minor diameter, and D p =2R p : principal di-ameter͒. The inverse aspect ratio ␦, defined as the ratio between minor and major diameter a / R, will play an important role in the expansion of the cavity mode field and the subse-PHYSICAL REVIEW A 76, 013823 ͑2007͒
doi:10.1103/physreva.76.013823 fatcat:okqn3mxvzjgo5nyzymllf773fm