Quality factor (Q)

For an optical resonator with resonance at frequency $f_0$ and full-width-half-maximum linewidth $\Delta f$ (or equivalently wavelength $\lambda_0$ and FWHM $\Delta \lambda$), the quality factor is

Higher $Q$ corresponds to a narrower resonance and longer photon lifetime in the cavity.

Two distinct $Q$ values are reported for coupled resonators:

Loaded $Q$ ($Q_L$) describes the measured resonance linewidth, combining intrinsic resonator losses with coupling losses to external waveguides. $Q_L$ is what is observed in the transmission spectrum.

Intrinsic $Q$ ($Q_i$) describes only the resonator's internal losses (propagation loss, bend loss, material absorption), excluding the coupling loss to the bus waveguide. $Q_i$ is the parameter relevant to material and fabrication quality.

The two are related through the coupling-limited $Q_c$:

For a ring resonator of group index $n_g$ and round-trip length $L$, the intrinsic $Q$ relates to propagation loss as

with $\alpha$ in units of 1/length. For SOI strip waveguides ($n_g \approx 4.3$, $\alpha \approx 0.5$ cm$^{-1} \approx 2$ dB/cm), this gives $Q_i \approx 1.7 \times 10^5$ at 1550 nm. For silicon nitride ($n_g \approx 2.0$, $\alpha \approx 0.01$ cm$^{-1} \approx 0.04$ dB/cm), $Q_i \approx 4 \times 10^6$.

Coupling regime classification depends on the relative magnitudes of $Q_i$ and $Q_c$:

Under-coupled and over-coupled rings produce identical transmission spectra at the same loaded $Q$. Disambiguation requires additional information from gap-sweep series, add-drop measurements, or phase-resolved detection. Methodology is covered in Loaded and Intrinsic Q Factor Extraction from Ring Resonator Transmission Spectra.