finesse (original) (raw)

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

Category: optical resonators

Related: cavitiesFabry–Perot interferometerssupermirrorsreference cavitiesbandwidthQ-factorfree spectral range

Units: (dimensionless)

Formula symbol: ($F$)

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DOI: 10.61835/hf3 Cite the article: BibTex BibLaTex plain textHTML Link to this page! LinkedIn

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Contents

What does Finesse Mean?

The finesse of an optical resonator (cavity) is a measure of 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.

Figure 1: Frequency-dependent transmissivity of a linear Fabry–Pérot interferometer with mirror reflectivities of 80%. The finesse is ≈ 14, and perfect mode matching is assumed.

Figure 2: Same as in Figure 2, but with higher mirror reflectivities of 90%. The finesse is ≈ 29.8.

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.

Figure 3: Finesse of a symmetric resonator as a function of the mirror reflectivity.

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 free spectral range. Instead, one must increase the finesse of the resonator.

A very high finesse (above 106) 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.

Frequently Asked Questions

This FAQ section was generated with AI based on the article content and has been reviewed by the article’s author (RP).

What is the finesse of an optical resonator?

The finesse of an optical resonator is a measure of how sharp its resonances are. It is defined as the free spectral range (the frequency spacing of resonances) divided by the bandwidth of these resonances. A high finesse indicates very sharp resonances.

What determines the finesse of a resonator?

The finesse is determined entirely by the resonator's round-trip losses and is independent of its length. It can be calculated from the fraction of optical power which is left after one round-trip.

How are high-finesse optical resonators used?

What is the relationship between finesse and the Q-factor?

Both finesse and the Q-factor characterize resonator losses. The Q-factor relates the resonance bandwidth to the absolute optical frequency, while the finesse relates it to the free spectral range. The Q-factor is the finesse multiplied by the ratio of the resonance frequency to the free spectral range.

Why might the measured finesse of a resonator appear lower than expected?

The apparent finesse can be reduced by the excitation of higher-order modes. If these modes have slightly different frequencies from the fundamental modes, they broaden the observed resonance peaks, resulting in a lower measured finesse.

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Bibliography

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