Quality factor (Q)

The quality factor, or Q, of a resonator measures how many optical cycles the light stored in it survives. For a resonance at frequency $f_0$, or vacuum wavelength ${\lambda }_0$, with a full width at half maximum $\mathrm{\Delta }f$, or $\mathrm{\Delta }\lambda$,

$$\begin{array}{rl}{\displaystyle Q}& {\displaystyle =\frac{f_0}{\mathrm{\Delta }f}=\frac{{\lambda }_0}{\mathrm{\Delta }\lambda }={\omega }_0{\tau }_p}\\ {\displaystyle }& {\displaystyle =2\pi \frac{\text{stored energy}}{\text{energy lost per cycle}},}\end{array}$$

where ${\omega }_0=2\pi f_0$ and ${\tau }_p$ is the photon lifetime, the time in which the stored energy falls to $1\mathrm{/}e$ once the input is removed. The energy therefore lasts $Q\mathrm{/}2\pi$ optical cycles: at 1550 nm a Q of ${10}^6$ means a linewidth of 193 MHz, or 1.55 pm, and a photon lifetime of 0.82 ns, about 160,000 cycles. The same number describes a Fabry-Perot cavity, a ring resonator, a photonic crystal nanocavity or a laser cavity, and the definitions agree closely whenever the light makes many round trips before it is lost. They part for very lossy cavities: a 300 µm laser diode chip with cleaved facets has, at 1550 nm, a Q of about $3\times {10}^3$ from its 2.4 ps photon lifetime, but about 20% less from the width of its broad fringes, which are not Lorentzian.

At 1550 nm:

Loaded, intrinsic and coupling Q

A resonator in use loses light in two ways: inside it, by absorption, scattering and radiation, and through the couplers or mirrors that connect it to the outside. Loss rates add, so the Q values combine as

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

where the loaded Q, $Q_L$, is the one measured from the linewidth, the intrinsic Q, $Q_i$, counts only the internal loss, and the coupling Q, $Q_c$, counts only the light leaving through the coupler. The intrinsic Q describes the material and the fabrication, and its parts can be separated in the same way, as $1\mathrm{/}{Q}_i=1\mathrm{/}{Q}_{\text{abs}}+1\mathrm{/}{Q}_{\text{scat}}+1\mathrm{/}{Q}_{\text{rad}}$. For a ring coupled to one bus waveguide, the power transmitted past the ring on resonance is

$$T_{\text{min}}={\left(\frac{Q_c-Q_i}{Q_c+Q_i}\right)}^2,$$

which defines three coupling regimes:

Measuring $Q_L$ and $T_{\text{min}}$ gives the intrinsic Q as

$$Q_i=\frac{2Q_L}{1\pm \sqrt{T_{\text{min}}}},$$

with the plus sign for an under-coupled ring and the minus sign for an over-coupled one. A resonance with $Q_L={10}^5$ and a 10 dB dip, $T_{\text{min}}=0.1$, gives $Q_i=1.52\times {10}^5$ if the ring is under-coupled and $2.92\times {10}^5$ if it is over-coupled, which for the silicon strip waveguide below means a loss of 4.8 or 2.5 dB/cm. The transmitted power alone cannot tell the two apart, since exchanging $Q_i$ and $Q_c$ leaves the power spectrum unchanged. A series of rings with different coupling gaps, an add-drop ring with a second bus waveguide, or a measurement of the phase across the resonance settles it, as described in Loaded and Intrinsic Q Factor Extraction from Ring Resonator Transmission Spectra. The Ring Resonator Explorer shows how the dip and the loaded Q follow from the coupling and the loss, and fits a measured scan on both branches.

Q and propagation loss

Light traveling at the group velocity $c\mathrm{/}{n}_g$ through a waveguide with a power attenuation coefficient $\alpha$ loses energy at the rate $\alpha c\mathrm{/}{n}_g$, so its lifetime is $n_g\mathrm{/}(\alpha c)$ and

$$Q_i=\frac{2\pi n_g}{{\lambda }_0\alpha },$$

where $n_g$ is the group index and $\alpha$ is in inverse length; 1 dB/cm is 23.0 m⁻¹. The size of the ring drops out. As long as the bends add no loss of their own, a ring's intrinsic Q measures the loss per unit length of its waveguide, whatever its circumference, while its finesse, the free spectral range divided by the linewidth, measures the loss per round trip and falls as the ring grows. At 1550 nm, $Q_i$ multiplied by the loss in dB/cm is $7.4\times {10}^5$ for a 500 × 220 nm silicon strip, whose computed group index is 4.18, and $3.7\times {10}^5$ for a 1600 × 800 nm silicon nitride strip, whose group index is 2.12:

A measured intrinsic Q is therefore also a loss measurement, one that needs no cutback series, although it lumps bend radiation, scattering at the coupler and absorption together with the loss of the straight waveguide. The Q to loss converter makes the conversion with platform presets for the group index, and Three ways to measure waveguide loss, compared sets the method against the cutback method and Fabry-Perot fringes. The same formula applies to a Fabry-Perot cavity with $\alpha$ replaced by the internal loss plus the mirror loss spread over the cavity length, which is how the laser chip above gets its Q.

Typical values

What Q sets

A resonator fed at critical coupling with an input power $P_{\text{in}}$ stores an energy $2P_{\text{in}}{Q}_L\mathrm{/}{\omega }_0$, so the intensity inside grows with Q. High-Q resonators are used for nonlinear optics for this reason: the pump power a Kerr microcomb needs to start falls as the square of Q. The stored light also takes a photon lifetime to build up or drain away, which limits the speed of resonant devices: the bandwidth of a microring modulator is capped near its linewidth, 22.9 GHz for a loaded Q of 10,000 at 1310 nm. In a laser the cavity Q fixes the photon lifetime and with it the gain needed to reach threshold. Q-switching is named for this: the cavity is held at low Q while the gain medium stores energy, then switched to high Q, and the energy leaves as a short, intense pulse.

Measuring Q

Q is usually measured from the width of a resonance in a transmission scan with a tunable laser, taking at least 20 points within the linewidth and fitting a Lorentzian, or the full transfer function when the resonance is not narrow compared with the free spectral range. Three effects distort the width. Above a Q of about ${10}^6$, where the linewidth at 1550 nm is below 1.6 pm, the swept laser's own wavelength axis becomes a significant source of error, and the scan is calibrated against the fringes of an unbalanced fiber interferometer or against modulation sidebands of known frequency spacing. At high power, absorbed light heats the resonator and drags the resonance along with the sweep, giving a triangular dip whose width depends on the direction of the sweep, so scans are taken at low power. In high-Q rings, backscattering from sidewall roughness couples the two directions of travel and splits a resonance into a doublet, which is fitted as a pair. For the highest Q values the photon lifetime is measured directly, by switching the input off and recording the decay of the output (cavity ring-down).

Common questions

What is the Q factor of a resonator?

The resonance frequency divided by the linewidth, $Q=f_0\mathrm{/}\mathrm{\Delta }f$, which equals 2π times the stored energy divided by the energy lost per optical cycle. A high Q means a narrow resonance and light that stays in the resonator for many cycles: at 1550 nm a Q of ${10}^6$ is a 1.55 pm linewidth and a 0.82 ns photon lifetime.

What is the difference between loaded and intrinsic Q?

The loaded Q is what the measured linewidth gives, and it includes the light that leaves through the coupler; the intrinsic Q counts only the loss inside the resonator. They are related by $1\mathrm{/}{Q}_L=1\mathrm{/}{Q}_i+1\mathrm{/}{Q}_c$, and at critical coupling the intrinsic Q is twice the loaded Q.

How is Q related to propagation loss?

For a ring or another resonator made of waveguide, $Q_i=2\pi n_g\mathrm{/}({\lambda }_0\alpha )$, with $\alpha$ the power attenuation coefficient, so the intrinsic Q is inversely proportional to the loss. At 1550 nm a silicon strip waveguide with 1 dB/cm of loss gives an intrinsic Q of $7.4\times {10}^5$.

What is the difference between Q and finesse?

Q compares the linewidth with the optical frequency and finesse compares it with the free spectral range, so $Q=m\mathcal{F}$, with $m$ the mode number. Finesse is set by the loss per round trip and Q by the loss per unit time, so a longer cavity with the same mirrors has a higher Q and the same finesse.

What is the Q-factor in optical communications?

A different quantity with the same name: the separation of the two levels of a binary signal divided by the sum of their noise, $Q=({\mu }_1-{\mu }_0)\mathrm{/}({\sigma }_1+{\sigma }_0)$, which sets the bit error ratio. A Q of 7 corresponds to a bit error ratio of about ${10}^{-12}$.

References: A. E. Siegman, Lasers (University Science Books, 1986). H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984). A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electronics Letters 36, 321 (2000). W. Bogaerts et al., "Silicon microring resonators," Laser & Photonics Reviews 6, 47 (2012). T. Asano et al., "Photonic crystal nanocavity with a Q factor exceeding eleven million," Optics Express 25, 1769 (2017). D. K. Armani et al., "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003). M. W. Puckett et al., "422 Million intrinsic quality factor planar integrated all-waveguide resonator with sub-MHz linewidth," Nature Communications 12, 934 (2021). T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Physical Review Letters 93, 083904 (2004).