# Finesse

Definition: the free spectral range divided by the FWHM width of the resonances of an optical resonator

German: Finesse

Units: (dimensionless)

Formula symbol: <$F$>

Author: Dr. Rüdiger Paschotta

The finesse of an optical resonator (cavity) is a measure for how narrow the resonances are in relation to their frequency distance: a high finesse means sharp resonances. It is defined as the free spectral range (i.e., the fundamental mode spacing) divided by the (full width at half-maximum) bandwidth of the resonances. It is fully determined by the resonator losses and is independent of the resonator length. It can be specified not only for Fabry–Pérot interferometers, but also for other types of optical resonators, and is often used in the context of resonators with sharp resonances, as used e.g. in optical frequency standards. It is not common to specify the finesse e.g. of a laser resonator.

For the calculation of the finesse, we assume that some light is circulating in the resonator while there is no incident field from outside the resonator. Some of the optical energy will be lost after each resonator round-trip. If a fraction <$\rho$> of the circulating power is left after one round-trip (i.e., a fraction <$1 - \rho$> of the power is lost), the finesse is

$$F = \frac{\pi }{{2\arcsin \left( {\frac{{1 - \sqrt \rho }}{{2\sqrt[4]{\rho }}}} \right)}} \approx \frac{\pi }{{1 - \sqrt \rho }} \approx \frac{{2\pi }}{{1 - \rho }}$$where the approximation holds for low round-trip losses (e.g., <10%), i.e., only for high finesse values. That is actually the regime in which the term *finesse* is mostly used.

## High-finesse Resonators for Spectral Analysis

An optical resonator with variable length can be used as a tunable frequency filter for spectral analysis: by measuring the optical power throughput as a function of resonator length (which is scanned e.g. with a piezo actuator behind one of the mirrors), one can obtain the optical spectrum, provided that it is limited to a region which is smaller than the free spectral range. Otherwise, multiple frequency components could be transmitted at the same time.

One does not arbitrarily increase the frequency resolution of such an optical spectrometer by choosing a long resonator length because that would lead to a too narrow respectable range. Instead, one must increase the finesse of the resonator.

A very high finesse (above 10^{6}) of a Fabry–Pérot resonator can be achieved by using dielectric supermirrors, which have a reflectance very close to 1 and exhibit very weak phase distortions. There are also high-finesse resonators of other types, for example microcavities based on whispering gallery modes, a compact kind of ring resonators.

## Apparently Reduced Finesse due to Higher-order Modes

Note that the apparent bandwidth of the resonances, observed e.g. by scanning the resonator length while observing the transmission with a single-frequency input wave, can appear to be increased due to the excitation of transverse modes with different orders. For a perfectly aligned confocal resonator, the frequencies of even higher-order modes are degenerate with frequencies of axial modes, so that this effect does not occur, but with some misalignment the modes are no longer perfectly degenerate. The apparent finesse can then be reduced.

## Relation of Finesse to the Q Factor

The finesse is related to the *Q* factor: the latter is the finesse times the resonance frequency divided by the free spectral range. Essentially, while the finesse relates the resonance bandwidth to the free spectral range, the Q factor relates it to the average optical frequency.

If one increases the resonator's round-trip length while keeping the power losses per round trip constant, the finesse will stay constant, while the Q factor will increase. The latter reflects that it will take more time for the internal optical energy to decay.

### Bibliography

[1] | E. D. Palik, H. Boukari, and R. W. Gammon, “Experimental study of the effect of surface defects on the finesse and contrast of a Fabry–Perot interferometer”, Appl. Opt. 35 (1), 38 (1996), DOI:10.1364/AO.35.000038 |

[2] | M. Suter and P. Dietiker, “Calculation of the finesse of an ideal Fabry–Perot resonator”, Appl. Opt. 53 (30), 7004 (2014), DOI:10.1364/AO.53.007004 |

[3] | N. Ismail et al., “Fabry-Pérot resonator: spectral line shapes, generic and related Airy distributions, linewidths, finesses, and performance at low or frequency-dependent reflectivity”, Opt. Express 24 (15), 16366 (2016), DOI:10.1364/OE.24.016366 |

See also: cavities, Fabry–Perot interferometers, supermirrors, reference cavities, bandwidth, Q factor, free spectral range

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