The Q-Factor of Optical Cavities

What Q means physically, how to compute it, and how mirror reflectance and cavity length determine photon lifetime.

A laser cavity with very high mirror reflectivity can store a photon for milliseconds. That sounds short until you realize a photon at 1064 nm bounces back and forth inside the cavity at the speed of light, so a millisecond is hundreds of thousands of round trips. The same photon, in a cavity with mirrors that are merely "good" (say 99% reflective), survives for tens of nanoseconds. The difference between "tens of nanoseconds" and "milliseconds" is the difference between a typical research-grade laser cavity and a LIGO arm. Both are described by the same number: the cavity Q-factor.

The Q-factor (quality factor) is a single dimensionless number that captures everything important about how well a resonator stores energy. It applies equally to a tuning fork, an LC circuit, a microwave cavity, an atomic transition, or an optical Fabry-Perot. Once you know Q, you know how narrow the resonance is, how long photons live, and how sharply the system rings when you kick it. This tutorial works through what Q means physically, how to compute it for an optical cavity from first principles, and how it ties to the mirrors you might design in our Thin-Film Coating Simulator.

If you have not read the Fabry-Perot Interferometers tutorial yet, that is a useful prerequisite. Most of the cavity formulas here come from there.

What Q Actually Means

Pick any resonator. Drive it gently at its resonant frequency and let it ring. Now let go. The amplitude decays exponentially. The energy stored in the resonator is proportional to amplitude squared, so it also decays exponentially:

where $\tau$ is the energy decay time constant. The Q-factor is defined as:

where $\omega =2\pi \nu$ is the angular frequency of the resonance. Equivalently:

These two definitions are the same. To see it, the energy lost in one optical cycle of duration $T=2\pi \mathrm{/}\omega$ is roughly $E\cdot T\mathrm{/}\tau =E\cdot 2\pi \mathrm{/}(\omega \tau )$, so the ratio of stored to per-cycle loss is $\omega \tau \mathrm{/}(2\pi )$, and $2\pi$ times that is $\omega \tau$. So Q is just the number of radians of oscillation that fit into one decay time. A high-Q oscillator rings for many cycles before its energy bleeds away. A low-Q oscillator damps out quickly.

There is a third equivalent statement, this time in the frequency domain. If you drive the resonator with a swept-frequency input and measure the response, you see a Lorentzian peak centered at the resonance frequency ${\nu }_0$ with full-width-at-half-maximum $\mathrm{\Delta }\nu$. The Q-factor and the linewidth are related by:

A high-Q resonator has a narrow linewidth. A low-Q one has a broad linewidth. These three definitions (energy storage, ringdown lifetime, frequency-domain linewidth) are all the same number, expressed three different ways. The Fourier transform of an exponential decay is a Lorentzian, and $\mathrm{\Delta }\nu \cdot \tau =1\mathrm{/}(2\pi )$, which makes $Q={\nu }_0\cdot 2\pi \cdot \tau =\omega \cdot \tau$. Same answer, three different ways of measuring it.

Optical Cavities: From Mirrors to Q

In an optical Fabry-Perot cavity, photons are confined between two parallel mirrors of reflectance $R_1$ and $R_2$, separated by distance $L$ (in a medium of refractive index $n$). On every round trip, the photon hits each mirror once and loses a fraction of its energy. The intracavity intensity decays as it bounces back and forth, and the decay rate is what determines the cavity Q.

Let me derive the photon lifetime carefully. The round-trip time inside the cavity is:

After one round trip, the intracavity intensity has been multiplied by $R_1\cdot R_2$ (energy reflectance, not field reflectance, because we are tracking power). For a symmetric cavity with $R_1=R_2=R$:

If we model this discrete decay as an exponential $I(t)=I_0\cdot \exp (-t\mathrm{/}{\tau }_p)$, we need:

This ${\tau }_p$ is the photon lifetime of the cavity: the 1/e decay time of the stored optical energy. For $R$ close to 1 (which is the regime that matters for high-Q cavities), the logarithm linearizes: $\mathrm{\mid }\ln R\mathrm{\mid }\approx (1-R)$, so:

And the cavity Q is just $\omega \cdot {\tau }_p=2\pi \nu \cdot {\tau }_p$.

You can also express Q using cavity quantities you already know from the Fabry-Perot tutorial: the free spectral range $\text{FSR}=c\mathrm{/}(2nL)$ and the finesse $\mathcal{F}=\pi \sqrt{R}\mathrm{/}(1-R)$. The cavity linewidth FWHM is $\text{FSR}\mathrm{/}\mathcal{F}$, and:

This is the cleanest form to remember. The Q is the optical frequency divided by the linewidth, and the linewidth comes from FSR and finesse. Both routes ($\omega {\tau }_p$ or $\nu \mathcal{F}\mathrm{/}\text{FSR}$) give the same answer.

For a really compact recipe in the high-R limit:

So Q scales linearly with cavity length, linearly with finesse, and inversely with wavelength. Long cavities and high finesse (i.e., highly reflective mirrors) make high Q. This is the entire game.

Worked Example 1: A Modest Cavity

Take a cavity with $R=99\mathrm{\%}$ mirrors, $L=10$ cm, and $\lambda =1064$ nm (a standard Nd:YAG wavelength, the kind of thing you might build in a teaching lab). Let me compute everything.

• Round-trip time: ${\tau }_{\text{rt}}=2\cdot 0.1\mathrm{/}(3\times {10}^8)=0.667$ ns
• FSR: $\text{FSR}=c\mathrm{/}(2L)=1.499$ GHz
• Finesse: $\mathcal{F}=\pi \sqrt{0.99}\mathrm{/}(1-0.99)=312.6$
• Linewidth: $\mathrm{\Delta }\nu =\text{FSR}\mathrm{/}\mathcal{F}=4.80$ MHz
• Photon lifetime: ${\tau }_p={\tau }_{\text{rt}}\mathrm{/}(2\mathrm{\mid }\ln 0.99\mathrm{\mid })=33.2$ ns. The photon survives for about 50 round trips before its energy decays by 1/e.
• Q-factor: $Q={\nu }_0\mathrm{/}\mathrm{\Delta }\nu =(2.82\times {10}^{14})\mathrm{/}(4.80\times {10}^6)=$ 5.88 × 10⁷

Cross-check via the photon lifetime: $\omega \cdot {\tau }_p=(2\pi \cdot 2.82\times {10}^{14})\cdot (33.2\times {10}^{-9})=5.88\times {10}^7$. Same answer.

So a 99% mirror cavity has $Q\approx 60$ million. That sounds like a lot. But modern dielectric coatings can do thousands of times better. Let us see what happens.

Worked Example 2: Pushing the Mirrors

Now keep the same cavity (10 cm, 1064 nm) but use much better mirrors: $R=99.99\mathrm{\%}$. This is achievable with a properly designed dielectric multilayer; we will return to that connection in a moment.

• Finesse: $\mathcal{F}=\pi \sqrt{0.9999}\mathrm{/}(1-0.9999)\approx$ 31,400
• Linewidth: $\mathrm{\Delta }\nu =1.499\text{GHz}\mathrm{/}\mathrm{31,400}=47.7$ kHz
• Photon lifetime: ${\tau }_p\approx {\tau }_{\text{rt}}\mathrm{/}(2\cdot 0.0001)=3.34$ μs. The photon survives for about 5,000 round trips.
• Q-factor: $Q=(2.82\times {10}^{14})\mathrm{/}(4.77\times {10}^4)=$ 5.91 × 10⁹

Improving R from 99% to 99.99% (a factor of 100 reduction in transmission) bought us a factor of 100 in Q. This is the key scaling: in the high-R limit, Q is inversely proportional to (1 − R). Every extra "9" you add to your mirror reflectivity gains you another factor of 10 in finesse and Q.

We are still not at the state of the art. With one more "9," the photon survives for an entire millisecond inside the cavity. That is the regime where you can literally watch the cavity ring down on a photodetector after the input is shuttered, and it is what enables one of the most sensitive trace-gas detection techniques ever invented.

Worked Example 3: An Ultrahigh-Finesse Cavity

How far can we push it? Modern ion-beam-sputtered dielectric mirrors achieve $R>99.999\mathrm{\%}$. Take $R=99.999\mathrm{\%}$, $L=50$ cm, $\lambda =1064$ nm:

• Finesse: $\mathcal{F}\approx$ 314,000
• Linewidth: $\mathrm{\Delta }\nu \approx$ 954 Hz
• Photon lifetime: ${\tau }_p\approx$ 167 μs (about 100,000 round trips)
• Q-factor: $Q\approx$ 3 × 10¹¹

This is the regime of cavity ringdown spectroscopy (CRDS), where you literally measure the photon lifetime by injecting a pulse and watching it decay. The 167 μs ringdown time is easily resolved with a fast photodetector, and the rate of decay (with and without an absorbing sample inside the cavity) is one of the most sensitive trace-gas detection techniques known. CRDS measures absorbances of 10⁻⁹ per pass routinely.

LIGO's 4 km arm cavities, by comparison, have a finesse of "only" ~225 (the input mirrors are deliberately partially transmissive to let the laser couple in and out), but the 4 km length wins back the Q. They reach $Q\approx 1.7\times {10}^{12}$. In the optical regime that is enormous, but it pales next to the Q of an atomic clock transition (10¹⁵ to 10¹⁷). The Q-factor lives on a logarithmic scale, and there is always something better.

Coupling Regimes: Loaded vs. Intrinsic Q

Suppose you build a cavity with $R=99.99\mathrm{\%}$ mirrors and you want to use it for trace-gas detection. You inject light through one mirror, the gas inside absorbs a tiny bit of it, and you measure the change in the cavity's transmitted (or reflected) signal. How tightly should you couple light into the cavity? Too loose, and not enough power makes it inside to interrogate the gas. Too tight, and the cavity stores almost no energy because the input mirror leaks too fast. There is a sweet spot, and the math that finds it is the difference between intrinsic and loaded Q.

Real cavities have two kinds of loss. First, there are mirror transmission losses you put in on purpose: you need photons to leak out so you can measure something. Second, there are intrinsic losses you wish you did not have: scattering at imperfect surfaces, absorption in the coating layers, transmission through the "back" mirror, diffraction loss for finite-size beams, parasitic absorption in the medium between the mirrors.

It is standard to separate these two loss channels and assign each its own Q:

• Intrinsic Q ($Q_0$): the Q the cavity would have if the only loss were the unavoidable internal losses. This is set by mirror absorption, scattering, and other internal mechanisms.
• Coupling Q ($Q_{\text{ext}}$): the Q the cavity would have if the only loss were the deliberate input/output coupling through one of the mirrors.

The total observed Q (called the loaded Q) combines them in parallel:

This is just the additivity of decay rates (and Q is inversely proportional to a decay rate), so two parallel loss channels add their inverse Q values.

There are three coupling regimes, distinguished by how the input coupling rate compares to the intrinsic loss rate:

• Under-coupled ($Q_{\text{ext}}>Q_0$): you are not letting enough light in. The intrinsic loss dominates. On resonance, most of the incoming light reflects off the input mirror because the cavity cannot suck in energy fast enough.
• Critically coupled ($Q_{\text{ext}}=Q_0$): the coupling rate exactly matches the intrinsic loss. On resonance, the reflected wave from the input mirror cancels exactly with the leakage of the intracavity field, and the on-resonance reflectance drops to zero. All incoming power is dissipated inside the cavity. This is the magic operating point for sensors and absorbers.
• Over-coupled ($Q_{\text{ext}}<Q_0$): you are letting too much light in. The cavity is dominated by the input coupling. On resonance, the intracavity field builds up high but the reflected wave still partially cancels.

At critical coupling, $Q_{\text{loaded}}=Q_0\mathrm{/}2$ (because the two terms in the parallel sum are equal). At critical coupling you measure half the intrinsic Q. To recover the true $Q_0$ from a critical-coupling linewidth measurement, you double it.

This is the fundamental tradeoff of every cavity-based experiment. Tightly coupled means you can see the resonance in reflection clearly but the cavity stores less energy. Weakly coupled means high intracavity buildup but the resonance is harder to interrogate. Critical coupling is the sweet spot for sensitive absorption measurements.

Connecting to Mirror Design

Where does R come from in the first place? For most laser cavities, the mirrors are dielectric multilayer stacks of alternating high- and low-index materials, the same kind you can design in our Thin-Film Coating Simulator. A symmetric quarter-wave stack of N high-index/low-index pairs on a substrate has a peak power reflectance:

with $n_H$ and $n_L$ the high and low refractive indices, $n_s$ the substrate index, and $n_0$ the ambient (air) index. This formula gives the intensity reflectance, not the field reflectance, so it is in the same units as the R that goes into the Q-factor formula above.

You do not have to remember this expression, because the simulator computes the exact reflectance for any layer stack including absorption and dispersion. But the scaling is what matters. Adding one more (HL) pair to the stack multiplies the residual transmission by $(n_L\mathrm{/}{n}_H{)}^2$. For a typical TiO₂/SiO₂ pair ($n_H\approx 2.4$, $n_L\approx 1.46$), that ratio is about 0.37, so each additional pair shrinks T by a factor of ~2.7. Twelve pairs gets you to $T\approx 4\times {10}^{-5}$ ($R\approx 99.996\mathrm{\%}$), and twenty pairs gets you to $T\approx {10}^{-8}$ in principle (though absorption sets a practical floor long before that).

For a deeper treatment of dielectric quarter-wave stacks (the HR mirrors that make ultra-high-Q cavities possible), see the High-Reflector Mirrors tutorial.

The workflow is:

1. Design the mirror in the Thin-Film Coating Simulator. Pick materials, pick a layer count, optimize for high R at your target wavelength.
2. Read off R at the design wavelength from the simulator's output.
3. Plug R into the Resonator Calculator to get the expected cavity finesse, photon lifetime, and Q.
4. Iterate. If your target Q requires R you cannot get with N pairs, add more pairs. If absorption losses dominate, try different materials.

This is exactly how cavity-builders work in practice. The mirror design and the cavity design are coupled: you cannot specify a target Q without knowing what R you can manufacture, and you cannot pick a mirror specification without knowing the Q you actually need.

Summary

The Q-factor of an optical cavity is one number that tells you everything about how well it stores light:

• $Q=\omega \cdot {\tau }_p$ ties Q to the photon lifetime.
• $Q=\nu \mathrm{/}\mathrm{\Delta }\nu$ ties Q to the resonance linewidth.
• $Q=\nu \cdot \mathcal{F}\mathrm{/}\text{FSR}$ ties Q to cavity-specific quantities (free spectral range and finesse).
• $Q\approx 2nL\cdot \mathcal{F}\mathrm{/}\lambda$ in the high-finesse limit, showing that long cavities and high mirror reflectivity dominate.

You can build this number out of mirrors you design in our coating simulator and a length you choose. From there, the application-relevant numbers (linewidth for laser stabilization, photon lifetime for ringdown spectroscopy, intracavity power buildup for nonlinear optics) follow from Q in one or two more lines. And if you have to share your cavity with the outside world (which you do, otherwise you can never measure anything), you separate Q into intrinsic and coupling parts and tune them against each other to land at critical coupling.

Q is the pivot variable of cavity optics. Once you internalize what Q is, the rest of the machinery (finesse, linewidth, photon lifetime, ringdown time) is just translation between equivalent ways of saying the same thing.

Going Further

To compute Q for your own cavity design, use our Resonator Calculator (it already exposes finesse, photon lifetime, and Q alongside the etalon and resonator-mode outputs, and is the live tool behind the embeds in this tutorial). To design the mirrors, use our Thin-Film Coating Simulator. The two tools are designed to be used together: design the coating, read off the peak R, plug it into the cavity calculator, iterate.

For the deeper theory:

• A. E. Siegman, Lasers (University Science Books, 1986). Chapters 11-12 on optical cavity fundamentals.
• H. Kogelnik and T. Li, "Laser beams and resonators," Applied Optics 5, 1550 (1966). Still the most cited reference.
• A. Yariv, Quantum Electronics (Wiley, 3rd ed. 1989). Chapter 7 on optical resonators with the cleanest treatment of loaded vs. intrinsic Q.
• K. J. Vahala, "Optical microcavities," Nature 424, 839 (2003). For where ultrahigh-Q cavities are headed (whispering-gallery resonators reach Q > 10¹⁰).

See how Q-factor applies to microring resonators → Ring Resonators in Photonics