Loaded and Intrinsic Q Factor Extraction from Ring Resonator Transmission Spectra

Scope

This article describes the procedure for extracting loaded and intrinsic quality factors $Q_L$ and $Q_i$ of an integrated ring resonator from its measured transmission spectrum. Coverage includes the all-pass and add-drop variants, the under-coupled / critically-coupled / over-coupled regime distinction, and the dominant sources of extraction error. Fano-shaped resonances and ring resonator design are outside scope.

Theory

All-pass ring resonator transmission

For a single bus waveguide coupled to a ring of round-trip length $L$, propagation loss $\alpha$, and field self-coupling coefficient $t$ (with $t^2 + \kappa^2 = 1$ assuming lossless coupling), the through-port transmission as a function of round-trip phase $\phi = \beta L$ is:

$$T(\phi) \;=\; \frac{a^2 - 2 a t \cos\phi + t^2}{1 - 2 a t \cos\phi + a^2 t^2},$$

where $a = \exp(-\alpha L / 2)$ is the single-pass field transmission of the ring.

At resonance ($\phi = 2\pi m$), the transmission reaches its minimum:

$$T_\text{min} \;=\; \left(\frac{a - t}{1 - a t}\right)^2.$$

The full width at half maximum (FWHM) of the resonance in wavelength is related to the loaded quality factor:

$$Q_L \;=\; \frac{\lambda_0}{\text{FWHM}},$$

where $\lambda_0$ is the resonance wavelength.

The loaded $Q_L$ combines the intrinsic loss of the ring (from propagation loss and bend loss) with the coupling loss to the bus waveguide. The intrinsic $Q_i$ describes only the ring loss:

$$\frac{1}{Q_L} \;=\; \frac{1}{Q_i} + \frac{1}{Q_c},$$

where $Q_c$ is the coupling-limited quality factor.

Coupling regimes

For a given measured pair $(Q_L, T_\text{min})$, the parameters $a$ and $t$ cannot be uniquely determined: two solutions exist, corresponding to the under-coupled regime ($t > a$) and the over-coupled regime ($t < a$). Both produce identical transmission spectra.

Under-coupled ($t > a$): coupling losses are smaller than intrinsic losses; the ring is dominated by its internal loss. $Q_i < Q_c$.

Critical coupling ($t = a$): coupling and intrinsic losses balance exactly. $T_\text{min} = 0$ (perfect extinction) and $Q_L = Q_i / 2 = Q_c / 2$.

Over-coupled ($t < a$): coupling losses are larger than intrinsic losses. Light bleeds out faster through the coupler than it dissipates inside. $Q_i > Q_c$.

Disambiguating the regime requires additional information beyond a single all-pass transmission measurement. Three methods are commonly used:

1. Multiple-ring measurement: fabricate rings with varying gap sizes (coupling strengths) and identify the trend. Stronger coupling drives $t \to 0$, so the regime is determined by which direction $t$ moves as the gap shrinks.
2. Add-drop variant: a second bus waveguide provides additional information (see below), enabling unique extraction from a single ring.
3. Phase response measurement: the phase of the transmission changes by $\pi$ at resonance for over-coupled rings, and remains $\sim 0$ for under-coupled rings. Requires interferometric measurement.

Add-drop ring transmission

For a ring coupled to two bus waveguides, the through-port and drop-port transmissions are:

$$T_\text{through}(\phi) \;=\; \frac{a^2 t_2^2 - 2 a t_1 t_2 \cos\phi + t_1^2}{1 - 2 a t_1 t_2 \cos\phi + (a t_1 t_2)^2}$$ $$T_\text{drop}(\phi) \;=\; \frac{a (1 - t_1^2)(1 - t_2^2)}{1 - 2 a t_1 t_2 \cos\phi + (a t_1 t_2)^2}$$

with two self-coupling coefficients $t_1$ and $t_2$ at the input and drop buses respectively.

The combined fit of through-port and drop-port spectra to a single resonance uniquely determines $t_1$, $t_2$, and $a$, eliminating the coupling-regime ambiguity that plagues all-pass measurements.

Equipment

For high-Q rings ($Q_L > 10^6$, FWHM $< 2$ pm at 1550 nm), tunable lasers with $\leq$ 0.1 pm tuning resolution are required. Step-and-measure mode is preferred over continuous sweep to avoid bandwidth limitations of the photodetector.

For low- and moderate-Q rings ($Q_L < 10^5$, FWHM $> 15$ pm), most commercial telecom tunable lasers (Agilent 81600, Santec TSL, etc.) at 1 pm step are sufficient.

Procedure

1. Determine the resonance and free spectral range (FSR)

Sweep the laser across a wide wavelength range (typically 5–10 nm) to identify the resonance wavelength $\lambda_0$ and the FSR. The FSR is:

$$\text{FSR} \;=\; \frac{\lambda_0^2}{n_g L},$$

where $n_g$ is the group index of the waveguide mode and $L$ is the round-trip length. For a typical SOI 220 nm strip waveguide ring with $L = 200~\mu\text{m}$, FSR $\approx 5$ nm at 1550 nm.

2. Sweep with adequate resolution

Set the sweep range to cover a single resonance with sufficient baseline on both sides — typically $\pm 5 \times$ FWHM. The number of samples within the FWHM should be $\geq 20$ for a clean fit.

For a ring with expected $Q_L \sim 10^5$ at 1550 nm (FWHM $\sim 15$ pm), a 200 pm sweep with 1 pm steps yields $\sim 15$ points within the FWHM — marginal. A 200 pm sweep with 0.5 pm steps yields $\sim 30$ points within the FWHM — comfortable.

3. Subtract baseline

The off-resonance transmission is set by the coupler and waveguide losses. For a stable measurement, this baseline appears as a flat region away from the resonance. Subtract the baseline (in linear power, not dB) so that the off-resonance transmission is normalized to unity.

For broadband measurements where the grating coupler response varies appreciably across the FSR, fit a low-order polynomial to the off-resonance baseline and divide it out before extracting the resonance.

4. Fit the resonance

For the all-pass variant, fit the resonance lineshape to the Lorentzian approximation (valid in the high-finesse limit, when FWHM $\ll$ FSR):

$$T(\lambda) \;=\; 1 - \frac{(1 - T_\text{min}) \cdot (\Gamma/2)^2}{(\lambda - \lambda_0)^2 + (\Gamma/2)^2},$$

where $\Gamma$ is the FWHM, $\lambda_0$ is the resonance wavelength, and $T_\text{min}$ is the minimum transmission. The fit has three free parameters: $\lambda_0$, $\Gamma$, and $T_\text{min}$.

For high-Q resonances with low FSR/FWHM ratios, the simple Lorentzian breaks down and the full transfer function must be used. Modern fitting routines (e.g., the SciPy curve_fit with an Airy-distribution-style transfer function) are standard.

Compute:

$$Q_L \;=\; \frac{\lambda_0}{\Gamma}.$$

5. Disambiguate the coupling regime

For all-pass rings, choose disambiguation method (see Theory) and apply. For add-drop rings, fit both through and drop port spectra simultaneously, with $t_1$, $t_2$, $a$, $\lambda_0$, and a baseline offset as free parameters.

6. Extract intrinsic Q

Once the regime is known, the field coefficients are determined:

$$\frac{1}{Q_L} \;=\; \frac{1}{Q_i} + \frac{1}{Q_c},$$

with $Q_i \propto 1/\ln(1/a)$ and $Q_c \propto 1/\ln(1/t)$.

For a ring of round-trip length $L$ and group index $n_g$:

$$Q_i \;=\; \frac{\pi n_g L}{\lambda_0 \ln(1/a)} \;\approx\; \frac{2 \pi n_g}{\lambda_0 \alpha},$$

where the approximation holds for $a$ near 1 and $\alpha$ is the propagation loss in 1/length. This relation enables propagation loss extraction from a measured $Q_i$:

$$\alpha \;=\; \frac{2 \pi n_g}{\lambda_0 Q_i}.$$

For $\lambda_0 = 1550$ nm, $n_g = 4.3$ (typical SOI strip), and $Q_i = 10^5$, this gives $\alpha = 0.18$ cm$^{-1}$ or $\sim 0.8$ dB/cm.

7. Report

Report:

• $Q_L$, $Q_i$, $Q_c$ for the resonance
• Resonance wavelength $\lambda_0$
• Minimum transmission $T_\text{min}$ (in dB)
• Coupling regime (under-coupled, critical, over-coupled)
• Fit residual standard deviation
• Method used to disambiguate regime

Worked example

A SOI 220 nm racetrack resonator with $L = 280~\mu\text{m}$ and $n_g = 4.30$ at 1550 nm is measured through grating couplers. A single sweep with 0.5 pm step gives the transmission spectrum around one resonance:

• Off-resonance transmission: $T_\text{base} = -16.5$ dBm
• Resonance wavelength: $\lambda_0 = 1549.832$ nm
• Resonance depth: $T_\text{min} - T_\text{base} = -22.3$ dB
• FWHM (after baseline normalization): $\Gamma = 12.4$ pm

Loaded Q:

$$Q_L \;=\; \frac{1549.832 \text{ nm}}{12.4 \text{ pm}} \;=\; 1.25 \times 10^5.$$

The deep resonance ($T_\text{min} = -22$ dB) indicates close-to-critical coupling. From a separate measurement on rings with varying coupling gap, this device is identified as slightly under-coupled. Using the field-coefficient relations:

$$a \;=\; 0.978, \qquad t \;=\; 0.992.$$

Intrinsic Q:

$$Q_i \;=\; \frac{\pi \cdot 4.30 \cdot 280 \times 10^{-6}}{1549.832 \times 10^{-9} \cdot \ln(1/0.978)} \;=\; 1.07 \times 10^5.$$

Corresponding propagation loss:

$$\alpha \;=\; \frac{2 \pi \cdot 4.30}{1549.832 \times 10^{-9} \cdot 1.07 \times 10^5} \;=\; 16.3 \text{ cm}^{-1} \;\approx\; 7.1 \text{ dB/cm}.$$

A propagation loss of 7 dB/cm is higher than expected for a standard SOI strip waveguide (1–3 dB/cm). The discrepancy suggests either bend loss contributions in the racetrack geometry, surface scattering from a higher-than-nominal sidewall roughness, or fit error from low resolution near the resonance peak.

Sources of extraction error

Inadequate wavelength resolution. Fewer than 10 points within the FWHM produces $Q_L$ values that systematically underestimate the true value, due to discretization of the lineshape peak. For high-Q rings, finer wavelength steps are required; if the laser cannot tune finely enough, the measured $Q_L$ should be reported as a lower bound.

Bandwidth-limited detection during continuous sweep. Photodetector bandwidth combined with sweep rate creates apparent broadening of the resonance. For a sweep rate $S$ (nm/s) and detector bandwidth $f$, the effective resolution floor is roughly $\Delta\lambda_\text{eff} \approx S/f$. For high-Q rings, step-and-measure operation eliminates this contribution.

Coupling drift during the sweep. Fiber position drift causes the off-resonance baseline to change during the sweep. The apparent $T_\text{min}$ then includes baseline variation, and the fitted lineshape is distorted. Standard mitigation: complete the sweep across a single resonance in $<1$ second; verify by repeated sweeps that baseline is reproducible to $<0.05$ dB.

Non-Lorentzian lineshape. The Lorentzian approximation breaks down for low-finesse rings (FWHM approaching FSR) and for rings with significant nonlinear losses at high power. For high-finesse rings, the Lorentzian is excellent; for low-finesse rings, the full Airy-distribution transfer function should be used.

Coupling-regime ambiguity not resolved. Reporting $Q_i$ without specifying the regime (or showing the disambiguation method) is a common error. Both regimes produce the same transmission spectrum but yield different $Q_i$ values; selecting the wrong regime can change extracted propagation loss by an order of magnitude.

Fano-shaped resonances from parasitic interferometers. Reflections at facets, fiber connectors, or other reflective surfaces produce Fabry–Pérot interference that distorts the ring resonance into a Fano lineshape. The Lorentzian fit fails for Fano-shaped resonances and produces large fit residuals; a Fano lineshape model must be used.

Mode hopping in tunable laser source. Some tunable lasers exhibit mode hops as wavelength is swept. These appear as discontinuous transmission jumps in the spectrum and can be mistaken for ring resonance features. Identifying and excluding mode-hop wavelength regions from the fit is required.

Validation

The Lorentzian fit residuals should be randomly distributed about zero. Visible asymmetry in the residuals indicates either a Fano-distorted lineshape or coupling drift during the sweep.

The extracted $Q_i$ implies a propagation loss $\alpha$ that should be consistent with the platform and waveguide cross-section. Loss values dramatically inconsistent with the platform suggest extraction error.

For rings designed with multiple coupling gaps on a single chip, the extracted $Q_c$ should vary smoothly with gap size, with $Q_i$ remaining approximately constant. A $Q_i$ that varies systematically with coupling gap indicates a fit or regime-identification error.

References

For the foundational ring resonator transmission analysis, see Heebner et al. (2008), Optical Microresonators. For modern fitting techniques and the add-drop disambiguation method, see Vermeulen et al. (2016) on unambiguous parameter extraction in add-drop ring resonators. For the practical context of ring resonator measurement in silicon photonics, see Bogaerts et al. (2012) on silicon microring resonators.