Fabry–Pérot interferometer

right|thumb|350px|[[Interference fringes, showing fine structure, from a Fabry–Pérot etalon. The source is a cooled deuterium lamp.]]

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

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

thumb|350px|Fabry–Pérot interferometer, using a pair of partially reflective, slightly wedged optical flats. The wedge angle is highly exaggerated in this illustration; only a fraction of a degree is actually necessary to avoid ghost fringes. Low-finesse versus high-finesse images correspond to mirror reflectivities of 4% (bare glass) and 95%.

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

thumb|right|350px|A commercial Fabry–Pérot device

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.

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.

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. 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.

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 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 <math> \ell </math>, homogeneously filled with a medium of refractive index <math> n </math>. Light is launched into the resonator under normal incidence. The round-trip time <math> t_{\rm RT} </math> of light travelling in the resonator with speed <math> c = c_0/n </math>, where <math> c_0 </math> is the speed of light in vacuum, and the free spectral range <math> \Delta \nu_{\rm FSR} </math> are given by

:<math> t_{\rm RT} = \frac{1}{\Delta \nu_{\rm FSR = \frac{2 \ell}{c}. </math>

The electric-field and intensity reflectivities <math> r_i </math> and <math> R_i </math>, respectively, at mirror <math> i </math> are

:<math> |r_i|^2 = R_i. </math>

If there are no other resonator losses, the decay of light intensity per round trip is quantified by the outcoupling decay-rate constant <math> 1 / \tau_{\rm out}, </math>

:<math> R_1 R_2 = e^{- t_{\rm RT} / \tau_{\rm out, </math>

and the photon-decay time <math> \tau_c </math> of the resonator is then given by

:<math> \frac{1}{\tau_c} = \frac{1}{\tau_{\rm out = \frac{-\ln{(R_1 R_2){t_{\rm RT. </math>

Resonance frequencies and spectral line shapes

With <math> \phi(\nu) </math> quantifying the single-pass phase shift that light exhibits when propagating from one mirror to the other, the round-trip phase shift at frequency <math> \nu </math> accumulates to

The decaying electric field at frequency <math> \nu_q </math> is represented by a damped harmonic oscillation with an initial amplitude of <math> E_{q,s} </math> and a decay-time constant of <math> 2 \tau_c </math>. In phasor notation, it can be expressed as 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 field <math> E_{\rm circ} </math> can be related to the field <math> E_{\rm laun} </math> that is launched into the resonator by

:<math> E_{\rm circ} = E_{\rm laun} + E_{\rm RT} = E_{\rm laun} + r_1 r_2 e^{-i 2 \phi} E_{\rm circ} \Rightarrow \frac{E_{\rm circ{E_{\rm laun = \frac{1}{1 - r_1 r_2 e^{-i 2 \phi. </math>

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, by tracing the infinite number of round trips that the incident electric field <math> E_\text{inc} </math> exhibits after entering the resonator and accumulating the electric field <math> E_\text{trans} </math> transmitted in all round trips. The field transmitted after the first propagation and the smaller and smaller fields transmitted after each consecutive propagation through the resonator are

:<math>\begin{align}

E_\text{trans,1} &= E_\text{inc}it_1 it_2 e^{-i \phi} = -E_\text{inc} t_1 t_2 e^{-i\phi}, \\

E_{\text{trans},m+1} &= E_{\text{trans},m}r_1 r_2 e^{-i2\phi},

\end{align}</math>

respectively. Exploiting

:<math> \sum_{m=0}^\infty x^m =

\frac{1}{1 - x} \Rightarrow E_\text{trans} =

\sum_{m=1}^\infty E_{\text{trans},m} =

E_\text{inc} \frac{- t_1 t_2 e^{-i \phi{1 - r_1 r_2 e^{-i 2 \phi

</math>

results in the same <math> E_\text{trans} / E_\text{inc} </math> as above, therefore the same Airy distribution <math> A_\text{trans}^{\prime} </math> derives. However, this approach is physically misleading, because it assumes that interference takes place between the outcoupled beams after mirror 2, outside the resonator, rather than the launched and circulating beams after mirror 1, inside the resonator. Since it is interference that modifies the spectral contents, the spectral intensity distribution inside the resonator would be the same as the incident spectral intensity distribution, and no resonance enhancement would occur inside the resonator.

Airy distribution as a sum of mode profiles

Physically, the Airy distribution is the sum of mode profiles of the longitudinal resonator modes.

:<math> 1 - L_{\rm RT} = e^{-\alpha_{\rm loss} 2 \ell } = e^{- t_{\rm RT} / \tau_{\rm loss. </math>

The additional loss shortens the photon-decay time <math> \tau_c </math> of the resonator:

: <math>\delta = \left( \frac{2 \pi}{\lambda} \right) 2n\ell \cos\theta. </math>

If both surfaces have a reflectance R, the transmittance function of the etalon is given by

: <math>T_e = \frac{(1 - R)^2}{1 - 2R\cos\delta + R^2} = \frac{1}{1 + F\sin^2\left(\frac{\delta}{2}\right)},</math>

where

: <math>F = \frac{4R}{(1 - R)^2}</math>

is the coefficient of finesse.

Maximum transmission (<math>T_e = 1</math>) occurs when the optical path length difference (<math>2 n l \cos\theta</math>) between each transmitted beam is an integer multiple of the wavelength. In the absence of absorption, the reflectance of the etalon Re is the complement of the transmittance, such that <math>T_e + R_e = 1</math>. The maximum reflectivity is given by

: <math>R_\max = 1 - \frac{1}{1 + F} = \frac{4R}{(1 + R)^2},</math>

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:

: <math>\Delta\lambda = \frac{\lambda_0^2}{2n_\mathrm{g} \ell \cos\theta + \lambda_0 } \approx \frac{\lambda_0^2}{2n_\mathrm{g} \ell \cos\theta},</math>

where λ0 is the central wavelength of the nearest transmission peak and <math>n_\mathrm{g}</math> is the group refractive index. The FSR is related to the full-width half-maximum, δλ, of any one transmission band by a quantity known as the finesse:

: <math>\mathcal{F} = \frac{\Delta\lambda}{\delta\lambda} = \frac{\pi}{2 \arcsin\left(\frac{1}{\sqrt F}\right)}.</math>

This is commonly approximated (for R > 0.5) by

: <math>\mathcal{F} \approx \frac{\pi \sqrt{F{2} = \frac{\pi R^\frac{1}{2{1 - R}.</math>

If the two mirrors are not equal, the finesse becomes

: <math>\mathcal{F} \approx \frac{\pi \left(R_1 R_2\right)^\frac{1}{4} }{1 - \left(R_1 R_2\right)^\frac{1}{2.</math>

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.

Two beams are shown in the diagram at the right, one of which (T0) is transmitted through the etalon, and the other of which (T1) is reflected twice before being transmitted. At each reflection, the amplitude is reduced by <math>\sqrt{R}</math>, while at each transmission through an interface the amplitude is reduced by <math>\sqrt{T}</math>. Assuming no absorption, conservation of energy requires T + R = 1. In the derivation below, n is the index of refraction inside the etalon, and n0 is that outside the etalon. It is presumed that n > n0. The incident amplitude at point a is taken to be one, and phasors are used to represent the amplitude of the radiation. The transmitted amplitude at point b will then be

: <math>t_0 = T\,e^{ik\ell/\cos\theta},</math>

where <math>k = 2\pi n/\lambda</math> is the wavenumber inside the etalon, and λ is the vacuum wavelength. At point c the transmitted amplitude will be

: <math>t'_1 = TR\,e^{3ik\ell/\cos\theta}.</math>

The total amplitude of both beams will be the sum of the amplitudes of the two beams measured along a line perpendicular to the direction of the beam. The amplitude t0 at point b can therefore be added to t1 retarded in phase by an amount <math>k_0 \ell_0</math>, where <math>k_0 = 2\pi n_0/\lambda</math> is the wavenumber outside of the etalon. Thus

: <math>t_1 = TR\,e^{\left(3ik\ell/\cos\theta\right) - ik_0 \ell_0},</math>

where ℓ0 is

: <math>\ell_0 = 2\ell\tan\theta\sin\theta_0.</math>

The phase difference between the two beams is

: <math>\delta = {2k\ell\over\cos\theta} - k_0 \ell_0.</math>

The relationship between θ and θ0 is given by Snell's law:

: <math>n \sin\theta = n_0 \sin\theta_0,</math>

so that the phase difference may be written as

: <math>\delta = 2k\ell\,\cos\theta.</math>

To within a constant multiplicative phase factor, the amplitude of the mth transmitted beam can be written as

: <math>t_m = TR^m e^{im\delta}.</math>

The total transmitted amplitude is the sum of all individual beams' amplitudes:

: <math>t = \sum_{m=0}^\infty t_m = T \sum_{m=0}^\infty R^m\,e^{im\delta}.</math>

The series is a geometric series, whose sum can be expressed analytically. The amplitude can be rewritten as

: <math>t = \frac{T}{1 - Re^{i\delta.</math>

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:

: <math>T_e = tt^* = \frac{T^2}{1 + R^2 - 2R\cos\delta}.</math>

For an asymmetrical cavity, that is, one with two different mirrors, the general form of the transmission function is

: <math>T_e = \frac{T_1 T_2}{1 + R_1 R_2 - 2 \sqrt{R_1 R_2} \cos\delta}.</math>

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.

Another expression for the transmission function was already derived in the description in frequency space as the infinite sum of all longitudinal mode profiles. Defining <math>\gamma = \ln\left(\frac{1}{R}\right)</math> the above expression may be written as

: <math>T_e = \frac{T^2}{1 - R^2}\left(\frac{\sinh\gamma}{\cosh\gamma - \cos\delta}\right).</math>

The second term is proportional to a wrapped Lorentzian distribution so that the transmission function may be written as a series of Lorentzian functions:

: <math>T_e = \frac{2\pi\,T^2}{1 - R^2}\,\sum_{\ell=-\infty}^\infty L(\delta - 2\pi\ell; \gamma),</math>