Finesse

Finesse measures how sharp a resonator's resonances are compared with their spacing:

$$\mathcal{F}=\frac{\mathrm{F}\mathrm{S}\mathrm{R}}{\mathrm{\Delta }{\nu }_{1\mathrm{/}2}}$$

where FSR is the free spectral range and $\mathrm{\Delta }{\nu }_{1\mathrm{/}2}$ the full width at half maximum of a resonance, both in the same units. A finesse of 300 means the cavity resolves 300 distinct frequencies within one period of its response, and it means that a photon survives roughly $\mathcal{F}\mathrm{/}2\pi$, about 50, round trips before it is lost. That second reading is the physical one: finesse is set entirely by the fractional loss per round trip, and unlike the quality factor it does not grow with the size of the resonator.

From the round-trip loss

With the round-trip amplitude factor $\rho =\sqrt{R_1{R}_2}{e}^{-{\alpha }_{i}L}$, which collects the two mirror reflectivities and the internal loss ${\alpha }_i$ over the length $L$, the finesse of a Fabry-Perot cavity is

$$\mathcal{F}=\frac{\pi \sqrt{\rho }}{1-\rho }$$

which reduces to $\pi \sqrt{R}\mathrm{/}(1-R)$ for equal lossless mirrors and to $\pi \mathrm{/}(1-R)$, or $2\pi$ divided by the round-trip power loss, when the loss is small. The same expression holds for a ring resonator with $\rho$ the amplitude transmission around the ring including the couplers. Internal loss enters on the same footing as mirror transmission, and since $e^{-{\alpha }_{i}L}$ caps $\rho$ however good the mirrors are, it caps the finesse: a 10 cm cavity in a medium with 0.001 dB/cm of loss cannot exceed a finesse of about 1,400 even with perfect mirrors, which is why the highest-finesse cavities are evacuated. The formula is the first-order one; the exact Airy linewidth differs from $\mathrm{F}\mathrm{S}\mathrm{R}\mathrm{/}\mathcal{F}$ by less than 0.001 % at $R=0.99$ and by 2 % at $R=0.5$, and the Fabry-Perot Cavity Calculator reports both.

Finesse, Q and photon lifetime

The three sharpness measures are the same information in different units. The quality factor counts linewidths in the optical frequency rather than in the FSR,

$$Q=\frac{{\nu }_0}{\mathrm{\Delta }\nu }=\mathcal{F}\frac{{\nu }_0}{\mathrm{F}\mathrm{S}\mathrm{R}}=m\mathcal{F}$$

with $m$ the longitudinal mode number, so a 10 cm cavity with a finesse of 314,000 at 1550 nm has $Q=4\times {10}^{10}$ while a silicon ring with a finesse of 300 and a mode number of 60 has $Q=18,000$. The photon lifetime is the linewidth turned into a time, ${\tau }_p=1\mathrm{/}(2\pi \mathrm{\Delta }\nu )=\mathcal{F}{T}_{rt}\mathrm{/}2\pi$ with $T_{rt}$ the round-trip time, which is the origin of the $\mathcal{F}\mathrm{/}2\pi$ round trips above. Finesse is the natural figure for filters and spectrometers, where the FSR is the period of interest; Q is the natural figure for lasers, sensors and nonlinear optics, where the absolute linewidth and the stored energy matter.

What limits it

Mirror transmission is the design choice, but the ceiling is set by everything else: mirror absorption and scatter, which for ion-beam-sputtered supermirrors sum to a few parts per million and allow finesses of a few hundred thousand; diffraction loss when the mirrors are too small or the cavity too close to instability; and in integrated resonators the propagation loss of the waveguide, which is why a silicon nitride ring at 1 dB/m reaches a finesse of tens of thousands while a silicon ring at 2 dB/cm stays in the hundreds. Measuring finesse is measuring the linewidth: a swept-laser transmission scan gives $\mathrm{\Delta }\nu$ and the FSR together, and cavity ring-down gives the photon lifetime directly for the highest finesses, where the linewidth is too narrow to scan.

References: A. E. Siegman, Lasers (University Science Books, 1986), chapter 11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), chapter 11. G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, "Measurement of ultralow losses in an optical interferometer," Optics Letters 17, 363 (1992).