Perturbative analytic theory of an ultrahigh-$Q$ toroidal microcavity

A perturbation theoretic approach is proposed as an efficient characterization tool for a tapered fiber coupled ultrahigh-quality factor $\left(Q\right)$ 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 ultrahigh-$Q$ 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.

Figure 1

A local toroidal coordinate system along with a corresponding boundary of the toroidal microcavity. The definitions of principal $(D_p=2R_p)$, major $(D=2R)$, and minor diameter $(d=2a)$ are illustrated along with an axis of rotation in the $z$ direction.

Figure 2

(Color online) (a) The ratio of the squared nontransverse to transverse electric fields for a fundamental quasi-TE mode (major diameter of the toroidal microcavity is $60\mathrm{\mu }\mathrm{m}$ and the resonance wavelength is chosen to be around $1550\mathrm{nm}$). (b) Resonance wavelength prediction by the perturbative analytic method and the full vectorial model.

Figure 3

(Color online) The mode index $(n_e={\beta }_e∕k)$ of the fundamental quasi-TE mode as a function of minor radius. The iteration method gives a slight variation in the mode index due to the change in actual resonance wavelengths $(\sim 1550\mathrm{nm})$ for different minor diameters.

Figure 4

(Color online) The normalized squared transverse electric field [fundamental quasi-TE mode with $(q,l)=(0,0)$] calculated from iterative perturbation theory for a toroidal microcavity with a minor (principal) diameter of $3\left(60\right)\mathrm{\mu }\mathrm{m}$ along with a theoretical prediction provided by a finite element method (FEM) simulation.

Figure 5

(Color online) The normalized squared transverse electric field [quasi-TE mode with $(q,l)=(1,0)$] calculated from iterative perturbation theory along with a theoretical prediction provided by a finite element method (FEM) simulation.

Figure 6

(Color online) Squared transverse electric fields for a toroidal microcavity with a minor (principal) diameter of $3\left(60\right)\mathrm{\mu }\mathrm{m}$.

Figure 7

(Color online) Left: resonance wavelengths for fundamental quasi-${\mathrm{TE}}_{00m}$ modes as a function of a minor diameter (principal diameter of toroidal microcavity is fixed at $60\mathrm{\mu }\mathrm{m}$). Resonance wavelengths are plotted for two adjacent azimuthal mode numbers $m=157$ and 158. Right: resonance wavelengths for quasi-${\mathrm{TE}}_{10m}$ modes.

Figure 8

(Color online) FSR calculated for the fundamental cavity mode $m=157$ and 158 as a function of the minor diameter (the principal diameter of the toroidal microcavity is fixed at $60\mathrm{\mu }\mathrm{m}$).

Figure 9

(Color online) Calculated mode volume $V$ of a fundamental $(q,l)=(0,0)$ mode of the toroidal microcavity for small inverse aspect ratio.

Figure 10

(Color online) Mode volume of higher-order modes calculated using the finite element method and the perturbative analytic calculations (the principal diameter of the toroidal microcavity is fixed at $60\mathrm{\mu }\mathrm{m}$).

Figure 11

Geometry and parameters of toroidal microcavity and tapered fiber used to derive the phase-matching condition.

Figure 12

(Color online) Phase-matched tapered fiber diameter as a function of the minor diameter for toroidal microcavities with 40, 60, and $80\mathrm{\mu }\mathrm{m}$ principal diameter.

Figure 13

(Color online) Radiative $Q$- factor for fundamental quasi-TE modes of toroidal microcavities with principal diameters of 40, 60, and $80\mathrm{\mu }\mathrm{m}$. The resonance wavelengths were chosen to be near $1550\mathrm{nm}$.