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In optics, a Fabry–Pérot interferometer (FPI) or etalon is an optical cavity made from two parallel reflecting surfaces (i.e.: thin mirrors). Optical waves can pass through the optical cavity only when they are in resonance with it. It is named after Charles Fabry and Alfred Perot, who developed the instrument in 1899.^[1][2][3] Etalon is from the French étalon, meaning "measuring gauge" or "standard".^[4]

Etalons are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light. Recent advances in fabrication technique allow the creation of very precise tunable Fabry–Pérot interferometers. The device is technically an interferometer when the distance between the two surfaces (and with it the resonance length) can be changed, and an etalon when the distance is fixed (however, the two terms are often used interchangeably).

Basic description

The heart of the Fabry–Pérot interferometer is a pair of partially reflective glass optical flats spaced micrometers to centimeters apart, with the reflective surfaces facing each other. (Alternatively, a Fabry–Pérot etalon uses a single plate with two parallel reflecting surfaces.) The flats in an interferometer are often made in a wedge shape to prevent the rear surfaces from producing interference fringes; the rear surfaces often also have an anti-reflective coating.

In a typical system, illumination is provided by a diffuse source set at the focal plane of a collimating lens. A focusing lens after the pair of flats would produce an inverted image of the source if the flats were not present; all light emitted from a point on the source is focused to a single point in the system's image plane. In the accompanying illustration, only one ray emitted from point A on the source is traced. As the ray passes through the paired flats, it is repeatedly reflected to produce multiple transmitted rays which are collected by the focusing lens and brought to point A' on the screen. The complete interference pattern takes the appearance of a set of concentric rings. The sharpness of the rings depends on the reflectivity of the flats. If the reflectivity is high, resulting in a high Q factor, monochromatic light produces a set of narrow bright rings against a dark background. A Fabry–Pérot interferometer with high Q is said to have high finesse.

Applications

Telecommunications

Telecommunications networks employing wavelength division multiplexing have add-drop multiplexers with banks of miniature tuned fused silica or diamond etalons. These are small iridescent cubes about 2 mm on a side, mounted in small high-precision racks. The materials are chosen to maintain stable mirror-to-mirror distances, and to keep stable frequencies even when the temperature varies. Diamond is preferred because it has greater heat conduction and still has a low coefficient of expansion. In 2005, some telecommunications equipment companies began using solid etalons that are themselves optical fibers. This eliminates most mounting, alignment and cooling difficulties.

Optical Instruments

Dichroic filters are made by depositing a series of etalonic layers on an optical surface by vapor deposition. These optical filters usually have more exact reflective and pass bands than absorptive filters. When properly designed, they run cooler than absorptive filters because they reflect unwanted wavelengths rather than absorbing them. Dichroic filters are widely used in optical equipment such as light sources, cameras, astronomical equipment, and laser systems.

Optical wavemeters and some optical spectrum analyzers use Fabry–Pérot interferometers with different free spectral ranges to determine the wavelength of light with great precision.

Laser resonators are often described as Fabry–Pérot resonators, although for many types of laser the reflectivity of one mirror is close to 100%, making it more similar to a Gires–Tournois interferometer. Semiconductor diode lasers sometimes use a true Fabry–Pérot geometry, due to the difficulty of coating the end facets of the chip. Quantum cascade lasers often employ Fabry–Pérot cavities to sustain lasing without the need for any facet coatings, due to the high gain of the active region.^[5]

Etalons are often placed inside the laser resonator when constructing single-mode lasers. Without an etalon, a laser will generally produce light over a wavelength range corresponding to a number of cavity modes, which are similar to Fabry–Pérot modes. Inserting an etalon into the laser cavity, with well-chosen finesse and free-spectral range, can suppress all cavity modes except for one, thus changing the operation of the laser from multi-mode to single-mode.

Stable Fabry–Pérot interferometers are often used to stabilize the frequency of light emitted by a laser (which often fluctuate due to mechanical vibrations or temperature changes) by means of locking it to a mode of the cavity. Many techniques exist to produce an error signal, such as the widely-used Pound–Drever–Hall technique.

Spectroscopy

Fabry–Pérot etalons can be used to prolong the interaction length in laser absorption spectrometry, particularly cavity ring-down, techniques. An etalon of increasing thickness can be used as a linear variable optical filter to achieve spectroscopy. It can be made incredibly small using thin films of nanometer thicknesses.^[6]

A Fabry–Pérot etalon can be used to make a spectrometer capable of observing the Zeeman effect, where the spectral lines are far too close together to distinguish with a normal spectrometer.

Astronomy

In astronomy an etalon is used to select a single atomic transition for imaging. The most common is the H-alpha line of the sun. The Ca-K line from the sun is also commonly imaged using etalons.

The methane sensor for Mars (MSM) aboard India's Mangalyaan is an example of a Fabry–Pérot instrument. It was the first Fabry–Pérot instrument in space when Mangalyaan launched.^[7] As it did not distinguish radiation absorbed by methane from radiation absorbed by carbon dioxide and other gases, it was later called an albedo mapper.^[8]

In gravitational wave detection, a Fabry–Pérot cavity is used to store photons for almost a millisecond while they bounce up and down between the mirrors. This increases the time a gravitational wave can interact with the light, which results in a better sensitivity at low frequencies. This principle is used by detectors such as LIGO and Virgo, which consist of a Michelson interferometer with a Fabry–Pérot cavity with a length of several kilometers in both arms. Smaller cavities, usually called mode cleaners, are used for spatial filtering and frequency stabilization of the main laser.^[9]

Theory

Resonator losses and outcoupled light

The spectral response of a Fabry–Pérot resonator is based on interference between the light launched into it and the light circulating in the resonator. Constructive interference occurs if the two beams are in phase, leading to resonant enhancement of light inside the resonator. If the two beams are out of phase, only a small portion of the launched light is stored inside the resonator. The stored, transmitted, and reflected light is spectrally modified compared to the incident light.

Assume a two-mirror Fabry–Pérot resonator of geometrical length ${\displaystyle \ell }$, homogeneously filled with a medium of refractive index ${\displaystyle n}$. Light is launched into the resonator under normal incidence. The round-trip time ${\displaystyle t_{\rm {RT}}}$ of light travelling in the resonator with speed ${\displaystyle c=c_{0}/n}$, where ${\displaystyle c_{0}}$ is the speed of light in vacuum, and the free spectral range ${\displaystyle \Delta \nu _{\rm {FSR}}}$ are given by

${\displaystyle t_{\rm {RT}}={\frac {1}{\Delta \nu _{\rm {FSR}}}}={\frac {2\ell }{c}}.}$

The electric-field and intensity reflectivities ${\displaystyle r_{i}}$ and ${\displaystyle R_{i}}$, respectively, at mirror ${\displaystyle i}$ are

${\displaystyle |r_{i}|^{2}=R_{i}.}$

If there are no other resonator losses, the decay of light intensity per round trip is quantified by the outcoupling decay-rate constant ${\displaystyle 1/\tau _{\rm {out}},}$

${\displaystyle R_{1}R_{2}=e^{-t_{\rm {RT}}/\tau _{\rm {out}}},}$

and the photon-decay time ${\displaystyle \tau _{c}}$ of the resonator is then given by^[10]

${\displaystyle {\frac {1}{\tau _{c}}}={\frac {1}{\tau _{\rm {out}}}}={\frac {-\ln {(R_{1}R_{2})}}{t_{\rm {RT}}}}.}$

Resonance frequencies and spectral line shapes

With ${\displaystyle \phi (\nu )}$ quantifying the single-pass phase shift that light exhibits when propagating from one mirror to the other, the round-trip phase shift at frequency ${\displaystyle \nu }$ accumulates to^[10]

${\displaystyle 2\phi (\nu )=2\pi \nu t_{\rm {RT}}.}$

Resonances occur at frequencies at which light exhibits constructive interference after one round trip. Each resonator mode with its mode index ${\displaystyle q}$, where ${\displaystyle q}$ is an integer in the interval ${\displaystyle }$, is associated with a resonance frequency ${\displaystyle \nu _{q}}$ and wavenumber ${\displaystyle k_{q}}$,

${\displaystyle \nu _{q}=q\Delta \nu _{\rm {FSR}}\Rightarrow k_{q}={\frac {2\pi q\Delta \nu _{\rm {FSR}}}{c}}.}$

Two modes with opposite values ${\displaystyle \pm q}$ and ${\displaystyle \pm k}$ of modal index and wavenumber, respectively, physically representing opposite propagation directions, occur at the same absolute value ${\displaystyle \left|\nu _{q}\right|}$ of frequency.^[11] Zdroj:https://en.wikipedia.org?pojem=Fabry–Pérot
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