*Q* Factor

Definition: a measure of the damping of resonator modes

German: Gütefaktor

Units: (dimensionless number)

Formula symbol: <$Q$>

Author: Dr. Rüdiger Paschotta

The *Q factor* (quality factor) of a resonator is a measure of the strength of the damping of its oscillations, or for the relative linewidth. The term was originally developed for electronic circuits, e.g. LC circuits, and for microwave cavities, also for mechanical resonators, but later also became common in the context of optical resonators.

There are actually two different common definitions of the <$Q$> factor of a resonator:

- Definition via energy storage: the <$Q$> factor is 2<$\pi$> times the ratio of the stored energy to the energy dissipated per oscillation cycle, or equivalently the ratio of the stored energy to the energy dissipated per radian of the oscillation. For a microwave or optical resonator, one oscillation cycle is understood as corresponding to the field oscillation period,
*not*the round-trip period (which may be much longer). - Definition via resonance bandwidth: the <$Q$> factor is the ratio of the resonance frequency <$\nu_0$> and the full width at half-maximum (FWHM) bandwidth <$\delta \nu$> of the resonance:

Both definitions are equivalent only in the limit of weakly damped oscillations, i.e. for high <$Q$> values. The term is mostly used in that regime.

*Q* Factor of an Optical Resonator

The <$Q$> factor of a resonator depends on the optical frequency <$\nu_0$>, the fractional power loss <$l$> per round trip, and the round-trip time <$T_\rm{rt}$>:

$$Q = \frac{{2\pi \: \nu_0 \: T_\rm{rt}}}{l}$$(assuming that <$l \ll 1$>).

For a resonator consisting of two mirrors with air (or vacuum) in between, the <$Q$> factor rises as the resonator length is increased because this decreases the energy loss per optical cycle. However, extremely high <$Q$> values (see below) are often achieved not by using very long resonators, but rather by strongly reducing the losses per round trip. For example, very high <$Q$> values are achieved with whispering gallery modes of tiny transparent spheres (see below).

## Important Relations

The <$Q$> factor of a resonator is related to various other quantities:

- The <$Q$> factor equals 2<$\pi$> times the exponential decay time of the stored energy times the optical frequency.
- The <$Q$> factor equals 2<$\pi$> times the number of oscillation periods required for the stored energy to decay to <$1/e$> (≈ 37%) of its initial value.
- The <$Q$> factor of an optical resonator equals the finesse times the optical frequency divided by the free spectral range.

## Intrinsic and Loaded Q Factor

The <$Q$> factor of an optical resonator is limited by optical losses, part of which can result from useful coupling to the external world – for example, through an output coupler mirror used for injecting light and probing the resonances. One may define the *intrinsic* <$Q$> factor as the value which results without the mentioned coupling; this is higher than the *loaded* <$Q$> factor obtained with the coupling. The inverse loaded <$Q$> factor is the sum of the inverse intrinsic <$Q$> factor and in addition due to the coupling.

For some kind of optical resonators, the coupling can be easily removed – for example, if it happens through frustrated total internal reflection over a gap which can be arbitrarily increased in width. That is the case for some whispering gallery mode microdisks, for example.

## High-*Q* Resonators

One possibility for achieving very high <$Q$> values is to use supermirrors with extremely low losses, suitable for ultra-high <$Q$> factors of the order of 10^{11}. Also, there are toroidal silica microcavities with dimensions of the order of 100 μm and <$Q$> factors well above 10^{8}, and silica microspheres with whispering gallery resonator modes exhibiting <$Q$> factors around 10^{10}.

High-<$Q$> optical resonators have various applications in fundamental research (e.g. in quantum optics) and also in telecommunications (e.g. as optical filters for separating WDM channels). Also, high-<$Q$> reference cavities are used in frequency metrology, e.g. for optical frequency standards. The <$Q$> factor then influences the precision with which the optical frequency of a laser can be stabilized to a cavity resonance.

## The *Q* Factor of an Oscillator

Sometimes, the term <$Q$> factor is applied to lasers and other kinds oscillators rather than to resonators. This requires additional careful thoughts, partly because a <$Q$> factor can then be defined in different ways:

- Considering the round-trip power losses of a laser resonator, there is also the laser gain, which in continuous-wave operation just compensates the losses. Taking into account that gain, one would arrive at effectively zero round-trip losses and there is an infinitely large <$Q$> factor. To avoid that problem, one may take the <$Q$> factor of the “cold” resonator, i.e., without laser gain. Many laser resonators exhibit a rather low <$Q$> factor; that is the case, for example, for most laser diodes. However, that parameter is not very relevant.
- One also take the definition based on the emission linewidth to get the <$Q$> factor as the ratio of the mean optical frequency to the bandwidth. That value, calculated for example for a laser oscillator, can be far higher than the cold-cavity value of the laser resonator. Similarly, an optical frequency standard can be operated with a very small linewidth, far below the natural linewidth of the used optical transition.

Generally, it should be recommended to use the term *Q factor* only for (passive) resonators, not for oscillators.

## Q switching

Although the term *Q factor* is not particularly common for laser resonators, it lead to the term Q switching, a method of pulse generation. When the <$Q$> factor of a laser resonator (based on its resonator losses only) is abruptly increased, an intense laser pulse (*giant pulse*) can generated. However, the magnitude of the <$Q$> factor during pulse generation is not particularly relevant for the obtained pulse properties; there is no need to maximize that value.

### Bibliography

[1] | S. L. McCall et al., “Whispering-gallery mode microdisk lasers”, Appl. Phys. Lett. 60 (3), 289 (1992), DOI: 10.1063/1.106688 |

[2] | M. L. Gorodetsky, A. A. Savchenkov and V. S. Ilchenko, “Ultimate <$Q$> of optical microsphere resonators”, Opt. Lett. 21 (7), 453 (1996), DOI:10.1364/OL.21.000453 |

[3] | D. K. Armani et al., “Ultra-high-Q toroid microcavity on a chip”, Nature 421, 925 (2003), DOI:10.1038/nature01371 |

[4] | C. R. Locke et al., “A simple technique for accurate and complete characterisation of a Fabry-Perot cavity”, Opt. Express 17 (24), 21935 (2009), DOI:10.1364/OE.17.021935 |

[5] | A. Muller et al., “Ultrahigh-finesse, low-mode-volume Fabry–Pérot microcavity”, Opt. Lett. 35 (13), 2293 (2010), DOI:10.1364/OL.35.002293 |

[6] | F. Vanier et al., “Intrinsic quality factor determination in whispering gallery mode microcavities using a single Stokes parameters measurement”, Opt. Express 19 (23), 23544 (2011), DOI:10.1364/OE.19.023544 |

[7] | L. Wu, “Greater than one billion Q factor for on-chip microresonators”, Opt. Lett. 45 (18), 5129 (2020), DOI:10.1364/OL.394940 |

See also: optical resonators, bandwidth, finesse, free spectral range, Q switching, reference cavities, optical frequency standards

## Questions and Comments from Users

2021-10-17

From my experiments with oscillators in electronics, I have found that Q is enhanced by the gain of the feedback amplifier up to the onset of oscillation. But once oscillation begins, the Q is not raised by the positive feedback and we do not see a bandwidth reduction, for instance.

The author's answer:

Indeed, you can consider the effective round-trip losses, which are reduced by the gain, and calculate a Q factor from that. If you crank up the amplifier gain, the Q factor becomes larger and larger, and isolation starts when Q becomes infinite.

I'm just not sure what you mean with Q during oscillation. Just redefining Q based on the emission linewidth is a problematic concept. The linewidth can be increased by all sorts of effects, not all related to the resonator.

2021-12-10

There is a relationship between Q-factor and finesse. Although the relationship seems obvious in a mathematical point of view, it does not make sense to me in an intuitive way. Is there a way to describe it in a more sensible way?

The author's answer:

It may help to consider that the Q factor refers to the loss of energy within one optical oscillation cycle, while the finesse refers to the losses per round trip. That explains the proportionality factor mentioned in the text.

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2020-12-02

Why is an isolated floating transmission line a high-Q oscillator?

The author's answer:

You probably mean an electrical transmission line with open or closed ends, which can act as a resonator. If the transmission losses and any losses at the ends are low, the Q factor is necessarily large: the initially stored energy will decay only slowly.