Fabry–Pérot interferometer
Fabry–Pérot interferometer
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 called an interferometer when the distance between the two surfaces (and with it the resonance length) can be changed, and 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 multiply 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 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.
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 can reflect unwanted wavelengths. 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.
Fabry–Pérot etalons can be used to prolong the interaction length in laser absorption spectrometry, particularly cavity ring-down, techniques.
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.
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.
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.
Theory
Resonator losses, outcoupled light, resonance frequencies, and spectral line shapes
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.
Fourier transformation of the electric field in time provides the electric field per unit frequency interval,
Each mode has a normalized spectral line shape per unit frequency interval given by
Generic Airy distribution: The internal resonance enhancement factor
The response of the Fabry-Pérot resonator to an electric field incident upon mirror 1 is described by several Airy distributions (named after the mathematician and astronomer George Biddell Airy) that quantify the light intensity in forward or backward propagation direction at different positions inside or outside the resonator with respect to either the launched or incident light intensity. The response of the Fabry-Pérot resonator is most easily derived by use of the circulating-field approach.[8] This approach assumes a steady state and relates the various electric fields to each other (see figure "Electric fields in a Fabry-Pérot resonator").
The generic Airy distribution, which considers solely the physical processes exhibited by light inside the resonator, then derives as the intensity circulating in the resonator relative to the intensity launched,[6]
Other Airy distributions
Once the internal resonance enhancement, the generic Airy distribution, is established, all other Airy distributions can be deduced by simple scaling factors.[6] Since the intensity launched into the resonator equals the transmitted fraction of the intensity incident upon mirror 1,
and the intensities transmitted through mirror 2, reflected at mirror 2, and transmitted through mirror 1 are the transmitted and reflected/transmitted fractions of the intensity circulating inside the resonator,
The index "emit" denotes Airy distributions that consider the sum of intensities emitted on both sides of the resonator.
It can be easily shown that in a Fabry-Pérot resonator, despite the occurrence of constructive and destructive interference, energy is conserved at all frequencies:
The external resonance enhancement factor (see figure "Resonance enhancement in a Fabry-Pérot resonator") is[6]
Usually light is transmitted through a Fabry-Pérot resonator. Therefore, an often applied Airy distribution is[6]
resulting in
respectively. Exploiting
Airy distribution as a sum of mode profiles
and then sums over the emitted mode profiles of all longitudinal modes[6]
Characterizing the Fabry-Pérot resonator: Lorentzian linewidth and finesse
The underlying Lorentzian lines can be resolved as long as the Taylor criterion is obeyed (see figure "The physical meaning of the Lorentzian finesse"). Consequently, one can define the Lorentzian finesse of a Fabry-Pérot resonator:[6]
Scanning the Fabry-Pérot resonator: Airy linewidth and finesse
![{\displaystyle \Delta \nu _{\rm {Airy}}=\Delta \nu _{\rm {FSR}}{\frac {2}{\pi }}\arcsin \left({\frac {1-{\sqrt {R_{1}R_{2}}}}{2{\sqrt[{4}]{R_{1}R_{2}}}}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d457f8ebf76a6238c058e1c126785b6422e5e44)
![{\displaystyle \Delta \nu _{\rm {Airy}}=\Delta \nu _{\rm {FSR}}\Rightarrow {\frac {1-{\sqrt {R_{1}R_{2}}}}{2{\sqrt[{4}]{R_{1}R_{2}}}}}=1\Rightarrow R_{1}R_{2}\approx 0.02944.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcbb818b7bf451a2f9c1e47c1db16460b72fb05)
![{\displaystyle {\mathcal {F}}_{\rm {Airy}}={\frac {\Delta \nu _{\rm {FSR}}}{\Delta \nu _{\rm {Airy}}}}={\frac {\pi }{2}}\left[\arcsin \left({\frac {1-{\sqrt {R_{1}R_{2}}}}{2{\sqrt[{4}]{R_{1}R_{2}}}}}\right)\right]^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/037eb36cd3332f19f6671f2d1fb6ea97d83426b3)
![{\displaystyle \Delta \nu _{\rm {Airy}}\approx \Delta \nu _{\rm {FSR}}{\frac {1}{\pi }}{\frac {1-{\sqrt {R_{1}R_{2}}}}{\sqrt[{4}]{R_{1}R_{2}}}}\Rightarrow {\mathcal {F}}_{\rm {Airy}}={\frac {\Delta \nu _{\rm {FSR}}}{\Delta \nu _{\rm {Airy}}}}\approx \pi {\frac {\sqrt[{4}]{R_{1}R_{2}}}{1-{\sqrt {R_{1}R_{2}}}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99690a8618e189381d01e21d52c5ff806af0bdfc)
Frequency-dependent mirror reflectivities
Fabry-Pérot resonator with intrinsic optical losses
![{\displaystyle {\frac {1}{\tau _{c}}}={\frac {1}{\tau _{\rm {out}}}}+{\frac {1}{\tau _{\rm {loss}}}}={\frac {-\ln {[R_{1}R_{2}(1-L_{\rm {RT}})]}}{t_{\rm {RT}}}}={\frac {-\ln {[R_{1}R_{2}]}}{t_{\rm {RT}}}}+c\alpha _{\rm {loss}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4e894ba1191388038a1ca640fe01aba1e0f59d)
The other Airy distributions can then be derived as above by additionally taking into account the propagation losses. Particularly, the transfer function with loss becomes
Description of the Fabry-Perot resonator in wavelength space
The varying transmission function of an etalon is caused by interference between the multiple reflections of light between the two reflecting surfaces. Constructive interference occurs if the transmitted beams are in phase, and this corresponds to a high-transmission peak of the etalon. If the transmitted beams are out-of-phase, destructive interference occurs and this corresponds to a transmission minimum. Whether the multiply reflected beams are in phase or not depends on the wavelength (λ) of the light (in vacuum), the angle the light travels through the etalon (θ), the thickness of the etalon (ℓ) and the refractive index of the material between the reflecting surfaces (n).
The phase difference between each successive transmitted pair (i.e. T2 and T1 in the diagram) is given by[10]
If both surfaces have a reflectance R, the transmittance function of the etalon is given by
where
is the coefficient of finesse.
and this occurs when the path-length difference is equal to half an odd multiple of the wavelength.
The wavelength separation between adjacent transmission peaks is called the free spectral range (FSR) of the etalon, Δλ, and is given by:
This is commonly approximated (for R > 0.5) by
If the two mirrors are not equal, the finesse becomes
Etalons with high finesse show sharper transmission peaks with lower minimum transmission coefficients. In the oblique incidence case, the finesse will depend on the polarization state of the beam, since the value of R, given by the Fresnel equations, is generally different for p and s polarizations.
where ℓ0 is
The phase difference between the two beams is
The relationship between θ and θ0 is given by Snell's law:
so that the phase difference may be written as
To within a constant multiplicative phase factor, the amplitude of the mth transmitted beam can be written as
The total transmitted amplitude is the sum of all individual beams' amplitudes:
The series is a geometric series, whose sum can be expressed analytically. The amplitude can be rewritten as
The intensity of the beam will be just t times its complex conjugate. Since the incident beam was assumed to have an intensity of one, this will also give the transmission function:
For an asymmetrical cavity, that is, one with two different mirrors, the general form of the transmission function is
A Fabry–Pérot interferometer differs from a Fabry–Pérot etalon in the fact that the distance ℓ between the plates can be tuned in order to change the wavelengths at which transmission peaks occur in the interferometer. Due to the angle dependence of the transmission, the peaks can also be shifted by rotating the etalon with respect to the beam.
The second term is proportional to a wrapped Lorentzian distribution so that the transmission function may be written as a series of Lorentzian functions:
where
See also
Lummer–Gehrcke interferometer
Gires–Tournois etalon
Atomic line filter
ARROW waveguide
Distributed Bragg reflector
Fiber Bragg grating
Optical microcavity
Thin-film interference