## Introduction

The Fabry–Perot Interferometer opens the route to superior precision spectroscopic measurements. Its extremely high spectral resolution allows it to measure the spectral line broadening and Doppler shift. Knowing these quantities, the particles group velocity and temperature can be evaluated for different species. These are important parameters determining the film growth mechanism.

The **Fabry–Perot Interferometer (FPI)** or **etalon** is a type of optical resonator or "cavity" and narrowband filter.
It consists of two partially transmitting mirrors facing each other. Depending on the application these mirrors can be flat or spherical and
the distance between them can range from micrometers to meters.
All Fabry-Perot designs share some common features but there are important differences which determine the right choice of the interferometer
for the particular application.

## How it works

The Fabry -Perot mirrors form an optical cavity in which successive reflections create multiple interference fringes. The simplest design is the flat mirror cavity, as shown in the figure below.

The characteristic behavior of a Fabry–Perot interferometer is exemplified by a simple case: A plane wave illuminates, at normal incidence, a FPI assembled from two flat, parallel mirrors spaced by $L$. Each mirror has reflectivity $R$ and scattering and absorption loss $T$. ($R$ and $T$ will sum to less than one because of absorption and scattering losses in the mirrors’ coatings.) The wave’s fractional transmission $\Psi$ through the etalon is given by the Airy function. \begin{equation} \label{Transm} \Psi = \frac{T^2}{1 + R^2 - 2R~cos\left [2\pi \dfrac{\nu }{c/2L} \right ]} \end{equation} where $\nu$ is the frequency and $c$ is the speed of light. $\Psi$ is periodic in frequency, repeating every free spectral range (FSR) \begin{equation} \Delta \nu = \frac{c}{2L} \end{equation} with maxima at $\nu_q = q~\Delta \nu$, for positive integers $q$. (Typically, $q \gg 1.$) For mirrors with low losses, $T \approx 1-R $, and equation (\ref{Transm}) then implies that the maximum value of $\Psi$ is almost 1.

If the mirrors also have high reflectivity ($R \geqslant 0.8$),$\Psi$ consists of narrow passbands ("transmission fringes"), centered at the $\nu_q$, amid broader frequency intervals having very low transmission. For such cases of high contrast between transmission peaks and valleys, a FPI’s spectral resolution $\delta \nu$ is conventionally taken to be the full-width-at-half-maximum of a fringe, and since $\delta \nu \ll \Delta \nu$, we can find a useful expression for $\delta \nu$ by expanding the cosine term in Eq. (\ref{Transm}) around any of the $\nu_q$. The result is \begin{equation} \delta \nu = \Delta \nu \frac{1-R}{\pi \sqrt R} \end{equation}

In general, the finesse $F$ of a FPI is defined as the ratio $\dfrac{\Delta \nu}{\delta \nu}$. For the present case, with high-reflectivity mirrors as the limiting factor \begin{equation} \label{Finess} F = \frac{\pi \sqrt R}{1 - R} \end{equation} Finesse is often taken as a figure of merit for a FPI, since, for a given $\Delta \nu$ "high" finesse implies "high" spectral resolution (i.e., "small" $\delta \nu$).

## How it works

Consider, for example, a FPI with $R = 0.95$ mirrors that are spaced by $5~cm$, and assume that
the interferometer’s finesse is reflectivity-limited. Then $F = 60$ and $\Delta \nu = 30~GHz$, hence, $\delta \nu = 50~MHz$.
At wavelength $\lambda = 600~nm$ this $\delta \nu$ corresponds to $\delta \lambda = 6 \cdot 10^{-5}~nm$, **a resolution that
is far better than that of a conventional diffraction-grating spectrometer or optical filter**.

Accompanying the FPI’s high resolution is its very limited FSR, corresponding to 0.0036 nm in this case. Thus, it picks up not just one single frequency but numerous frequencies separated by the interval $\Delta \nu$.

Thus a single FPI is not useful for spectroscopy of broadband sources. It would need to be filtered with a lower-resolution device, such as a grating spectrometer.

Although Eq.(\ref{Finess}) suggests that the $F$ can be very large for $R \approx 1$, other effects enter to limit the resolution (hence, finesse).

- First, plane waves are an idealization, and any monochromatic beam of finite transverse extent will consist of an angular spread of plane waves that experience differing transmissions through the parallel-plate FPI as a function of incidence angle. It becomes evident when Eq.(\ref{Transm}) is generalized for arbitrary incidence angles. The net effect is to broaden the fringes and, hence, degrade the spectral resolution (and $F$).
- Second, enlarging the beam size to reduce its angular content and, thus combat the problem just cited yields diminishing returns when the corresponding mirror apertures become so large that they can’t be polished sufficiently flat; this nonflat- ness of practical interferometer optics also reduces $F$. Limiting values of $F$ for flat–flat FPIs are about 100.

Fabry–Perot interferometers incorporating at least one concave spherical mirror provide a route to much higher finesse.In contrast with the mirrors in flat–flat FPIs, which let the incident beam continue its diffractive spreading as it bounces back and forth between them, concave spherical mirrors keep refocussing the beam. If a mode-matched beam is injected into such an interferometer, this refocusing is just right to compensate for diffraction. Thus the beam retains its small cross section during multiple passes, and mirrors that are only "locally smooth" suffice to give excellent performance. Further, multilayer dielectric coatings of high reflectivity can be deposited on such mirrors without greatly affecting their surface figures, in contrast to the mirrors for a flat–flat FPI, which may be permanently warped ~degrading the finesse! by exposure to the elevated temperatures needed for adding the coatings. Finally, mirror parallelism is critical for the performance of a flat–flat FPI, whereas small "misalignments" of the mirrors in a spherical-mirror FPI still permit achievement of high finesse—such misalignments simply displace and/or reorient the interferometer’s optical axis. The input beam’s steering can be adjusted to compensate.

The modert Fabry-Perot Interferometer with piezoelectric tuning is pictured in the figure below.

Fabry-Perot interferometer typically**features an internal housing made of thermally stable Invar to eliminate misalignment due to temperature**fluctuations. The transmission frequencies are tuned by adjusting the length of the resonator cavity using piezoelectric transducers, as shown in the diagram below.

## Throughput and Étendue

An advantage of the interferometers over other types of high resolution spectrometers is their efficiency, both in the transmission and throughput,
or light-gathering power.
For small apertures or perfectly flat and parallel mirrors, the transmission on the peak of a fringe is
\begin{equation}
\label{Tr}
\Psi_{max} = \left( 1 - \frac{T}{1 - R} \right)^2
\end{equation}
depends on $T$, the scattering and absorption loss at the mirrors.
For modern multilayer dielectric mirrors $T \leq 0.2 \% $. Consequently, mirror reflectivity $R$ as high as $98 \%$
can yield throughput close to $80 \%$ over a small aperture. The étendue for a plane mirror Fabry-Perot interferometer is
\begin{equation}
\label{etendue}
U = \Omega \cdot A_m = \dfrac{\pi~ D^2~ \lambda}{4~d~ F}
\end{equation}
All the radiation at wavelength $\lambda$ within a solid angle $\Omega$ subtended at a mirror aperture $A_m$ can be transmitted in the bandpass
defined by the instrumental finesse $F$. The above formulae apply to the Fabry-Perot interferometer using plane mirrors.
Similar Formulae exist for spherical mirror interferometers.
The most common interferometer of this type is the confocal design, where identical concave mirrors are spaced by precisely their radius of curvature.
For this case the free spectral range (in wave-number units $1/m$) $\Delta \nu = \dfrac{1}{4~n~ L}$, **or half of a plane-mirror system**.

## Choosing the Right Fabry-Perot

The best Fabry-Perot interferometer for a particular application depends on many factors, including size, stability, tunability, free spectral range, resolution, light-gathering power, and price. The distinguishing features of the various types of Fabry-Perot systems are outlined below.or*Solid etalon*are stable and compact, making them ideal for wavelength filtering, frequency calibration, coherence extension and intra-cavity mode selection in lasers. Solid etalons are made from a piece of a optically homogeneous materials such as fused quartz. Opposite faces are polished flat and parallel, and coated to any desired reflectivity. In a fixed-air-gap etalon, two mirrors are bonded to a solid spacer element.*fixed-air-gap etalons*Both types are highly stable mechanically, but solid etalons are more sensitive to temperature changes. A solid etalon is best used in a thermally controlled housing where it can be temperature tuned or stabilized. Fixed-air-gap etalons are more stable thermally and, unlike solid etalons, they can be pressure tuned. Both types allow no mechanical variability in spacing; the right spacing must be preselected for a specific application.

The simplest way to tune either etalons is by tilting. This is a good technique provided that tilt angle is not so large as to degrade the finesse. These etalons are difficult to manufacture with very flat and parallel surfaces, especially with large mirror spacing. They are best suited for optical systems with small diameter laser beams.

are similar to fixed-air-gap etalons, except the spacing is established by a mechanically adjustable frame in which the etalon plates are mounted. While adjustable mirror spacing is an advantage, this design is less stable - both mechanically and thermally than the bonded etalon. Applications are similar.*Variable spacing air gap etalons*

*Piezoelectric mirror control* is available for both fixed-air-gap and variable-spacing etalons.
In the former the piezoelectric elements are carefully matched in length and cemented directly to Fabry-Perot mirrors.
The later consists of a housing with a build-in piezoelectric assembly that supports the Fabry-Perot mirrors.
Some mechanical means of adjusting the mirror spacing and aligning the mirrors is provided.
In many cases, this difficulty in initial alignment is not important because once the adjustment is set the range
of piezoelectric control is sufficient to subsequently optimize the interferometer mirror alignment.

The small size and simple integrity of piezoelectric etalons enhances thermal and mechanical stability. Because these etalons are electronically tunable they can be used with active stabilisation system. Fixed-air-gap etalons with piezoelectric control have been built with capacitance displacement transducers that can be used for automatic alignment and cavity stabilization.

Applications for piezoelectrically tunable etalons include spectral analysis, laser tuning, active optical filtering and spectroscopy — all examples where it is not essential to have large spacing with full cavity adjustment, high finesse, or high étendue.

**have two identical concave mirrors spaced precisely at their common radius of curvature. Each mirror images the other back upon itself so that any paraxial ray entering the interferometer is superimposed upon itself after four reflections, resulting in a very high étendue. Because the mirrors are spherical, the requirements for a precise parallel alignment is greatly relaxed and only axial piezoelectric tuning is necessary. A typical confocal interferometer has cavity spacing of $50~cm$ and can resolve $1~MHz$.**

*Confocal etalons*Confocal interferometers are commonly called spectrum analysers when used for laser mode analysis. Because mirror alignment is not critical, they are easier to temperature stabilize than other high resolution Fabry-Perot interferometers. Thus they are often used as a passive reference cavities to stabilise laser frequency and to calibrate frequency of tunable laser. The high étendue also offers an advantage for high resolution spectroscopy of diffusive light source.

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