Fabry-Pérot etalons in solar astronomy. A review

During the last decades, the use of Fabry-Pérot etalons as filtergraphs has become frequent in solar instruments. The main reason is that they stand out for offering quasi-monochromatic, two-dimensional fields of view much higher than those provided by conventional slit-based spectrographs in a given time interval. Unfortunately, they also present several drawbacks. The number of etalons employed, the chosen way to illuminate them and the material they are made of have a large influence on the performance of the instrument. In this work we review and discuss the main results obtained by some of the most relevant studies in the design of etalon-based instruments. We present the general properties of etalons and their particularities when employed in solar instruments. We examine the (common) use of systems of several etalons to increase the free spectral range and to narrow down the filter transmission width. We compare the advantages and drawbacks of the two most common configurations —collimated and telecentric— paying special attention to their limitations. Finally, we also inspect the properties of crystalline etalons and their use in solar astronomy.


Introduction: why etalons?
Accurate mappings of the solar magnetic field are of paramount importance to address some of the fundamental questions that remain unanswered about our star. The origin of the solar dynamo responsible for its structure and periodic cycles, the way different layers of the solar atmosphere are magnetically coupled, or how the heating of the chromosphere and corona is produced are a few of those questions. Solar magnetism cannot be measured in situ, though. The combination of high spectral resolution and large polarimetric sensitivity in remote sensing observations is mandatory. The retrieval of the strength, orientation, structure, and evolution of the solar vector magnetic field is carried out with spectropolarimeters that measure the traces that isotropy-breaking, physical mechanisms like the Zeeman and Hanle effects, scattering, or the like leave in the polarization state of spectral lines formed in the solar atmosphere.
Spectropolarimeters are typically designed in two flavors, namely, tunable imaging spectropolarimeters, often (improperly) called magnetographs, and spectrograph-slitbased polarimeters. Here we concentrate on spectropolarimeters of the first type. Magnetographs image the solar surface in linear combinations of the Stokes parameters at given wavelengths along one or several spectral lines that are sensitive to the magnetic field. The spectropolarimetric properties of the emitted light encode information about the solar thermodynamic, magnetic, and dynamic properties, which can be inferred later through sophisticated inversions of the radiative transfer equation (see Del Toro Iniesta and Ruiz Cobo 2016, for a review). This is why we say they are improperly called magnetographs: the original maps they produce are not of the solar magnetic field. Magnetograms are obtained after post-facto data reduction. We shall continue using the term hereafter, though.
Modern magnetographs aim at providing maps of the magnetic field with sub-arcsec angular resolutions in order to resolve structures on the solar surface as small as several ∼ 10's of kilometers while sensing faint magnetic signals of about a few Gauss in times shorter than the typi-cal timescales of magnetic structures. To this purpose, fast diffraction-limited observations with a temporal resolution of ≤ 1 min, challenging polarimetric sensitivities of ∼ 10 −3 (Stokes Q, U , and V noise in units of the continuum intensity) and resolving powers of ∼ 10 5 or better are typically requested (e.g., Martínez Pillet et al. 2011). Hence, solar instruments must demonstrate not only an excellent performance in terms of image quality, but they must also provide signal-to-noise ratios higher than 10 3 while achieving an accurate and rapid scanning of the spectral lines of interest.
The wavelength tuning has been performed historically either with slit-based spectrographs or through narrow-band tunable filters. 1 Examples of the former technology are the Tenerife Infrared Polarimeter at the VTT (Martínez Pillet et al. 1999), the Diffraction-Limited Spectropolarimeter and SPINOR at the NSO Dunn Solar Telescope (Tritschler et al. 2007;Socas-Navarro et al. 2006), the Hinode Spectropolarimeter SP (Lites et al. 2001), or the multiline polarimetric mode of THEMIS (Molodij and Rayrole 2003). Their main advantage over most bidimensional imaging filtergraphs is that they capture a full unidimensional spectrum of light with a large resolving power (150,000 − 250,000), and they do so in a single exposure. In exchange, they sacrifice spatial information as they do not record the whole solar scene in a shot, but they need to scan over one spatial dimension to get an image of the solar surface ( Fig. 1, right). The timeconsuming scanning of the solar surface with such an accurate wavelength sampling limits the study of fast solar events with reasonable fields of view and a large optical resolution.
Meanwhile, magnetographs based on tunable filters enable an accurate and rapid imaging of the solar scene, but only at some selected (discrete) wavelengths along a spectral line (Fig. 1, left) and typically with lower resolving powers than slit-based spectrographs (∼ 50,000 − 150,000). 2 One important advantage of their fast imaging capabilities is that post-facto image reconstruction techniques are easier to implement. The most powerful image-processing techniques used nowadays in solar filtergraphs are possibly Phase Diversity (Gonsalves 1982;Paxman et al. 1992), Speckle interferometry (von der Lühe 1993), and (Multi-Object) Multi-Frame Blind Deconvolution (van Noort et al. 2005). These methods provide precise information of the wavefront error introduced by the instrument and allow for realistic (diffraction-limited) reconstructions with minimal artifacts. Fig. 1 Schematic layout of the working principle of a narrow-band filtergraph (left) and of a slit-based spectrograph (right). The two spatial dimensions of the solar scene (x and y) are captured simultaneously in the former, while the wavelength dimension (λ) is scanned sequentially in a time interval t . In the latter, an spectrum over one dimension (y) is obtained in each frame, while the scanning is carried out through the other dimension (x). Figure adapted from Iglesias and Feller (2019) with permission. Licensed under CC BY 4.0 Thus, they enable instruments based on tunable filters for the study of fast events with the highest optical quality.
On the other side, image restoration techniques in grating-based spectrographs have proven to be difficult to implement due to the shortage of detailed spatial information in every single exposure. Different solutions to overcome this issue are still in development and need further investigation (e.g., Quintero Noda et al. 2015), although some novel studies have shown that achieving (near) diffractionlimited restorations is possible when a complementary slitjaw camera is used to measure the degradation of the wavefront on the slit of the spectrograph through conventional image reconstruction techniques (van Noort 2017).
Another advantage of tunable filters is that they can be used together with dual-beam polarimetric techniques (e.g., Martínez Pillet et al. 2011;Scharmer et al. 2008). Dualbeam polarimetry consists in splitting the orthogonal polarization components of incident light with a polarizing beam splitter; each one being recorded in a different detector (or in different parts of a single detector). Observing simultaneously two perpendicular states of polarization makes it possible to reduce induced cross-talks between different Stokes parameters that appear as a result of differential motions between pairs consecutive images arising from jitter or seeing effects (Lites, 1987;Collados, 1999;Del Toro Iniesta, 2003). 3 The only requirement for the use of this technique is that the narrow-band filter needs to be placed before the 3 Polarimetry is based on differential imaging techniques of the modulated observed object. Therefore, differences between consecutive images are translated into polarimetric signals. If such differences are originated by a shift of one image with respect to the other, then the inferred Stokes vector will be corrupted by an artificial contamination among different Stokes components. Such a displacement can be origi- polarimetric analyzer (beam splitter). Otherwise, two filters would be needed, one for each channel, with a consequent increasing of the complexity in the design an operation of the instrument. Among the tunable filters, Fabry-Pérot interferometers (also called "etalons") are the most common choice in solar post-focus instruments (Iglesias and Feller 2019). Etalons were first developed in 1897 by Charles Fabry and Alfred Pérot (Pérot and Fabry 1897). Its applications to spectroscopy and astronomy were rapidly found (Perot and Fabry 1899;Fabry and Perot 1902), but the optical properties of these devices were not studied in detail until later by Chabbal (1953), Jacquinot (1960), Ramsay (1969), andHernandez (1988), among others. For a comprehensive historical description of the Fabry-Pérot interferometer and its applications, we refer the reader to Vaughan (1989).
Some examples of instruments based on Fabry-Pérot etalons are the Italian Panoramic Monochromator (IPM) at THEMIS (Bonaccini et al. 1989, and references therein), the TESOS spectrometer at the VTT (Kentischer et al. 1998), the Interferometric Bidimensional Spectrometer (IBIS) at the Dunn Solar Telescope of the Sacramento Peak Observatory (Cavallini 1998), the CRisp Imaging SpectroPolarimeter instrument (CRISP) at the Swedish 1-m Solar Telescope (Scharmer et al. 2008), the Imaging Magnetograph eXperiment (IMaX) instrument (Martínez Pillet et al. 2011) aboard the SUNRISE balloon observatory (Barthol et al. 2011), the nated as a consequence of the atmospheric seeing and/or of mechanical vibrations of the telescope (jitter) that induce instabilities on the image position.
GFPI at GREGOR (Puschmann et al. 2013), the PHI instrument (Solanki et al. 2020) on board the Solar Orbiter mission (Müller et al. 2020) and the future Visible Tunable Filter (VTF) (Schmidt et al. 2016) at DKIST (Rimmele et al. 2020). Some of the basic features of the mentioned etalonbased instruments are displayed in Table 1. Their success in solar instrumentation comes from their easiness of both use and interpretation of the measured data (unlike, for instance, Michelson interferometers), as well as for their high spectral resolution. They also show transmissions close to 100% and are almost insensitive to the polarization of the incident light if filled with air (Doerr et al. 2008), contrary to Lyot filters.
In this work we describe the general performance of etalons and their possible configurations on solar instruments through a review of most of the studies on these topics so far. We begin in Sect. 2 with an overview of the general properties of etalons and the way series of etalons are usually combined to improve their free spectral range. We continue in Sect. 3 with a review of the setups generally employed in solar instruments: collimated and telecentric. In Sect. 4 we examine the main advantages and drawbacks of crystalline etalons, as well as their potentialities in solar instruments. We finally present some conclusions in Sect. 5.

Spectral properties
Fabry-Pérot interferometers are basically resonant optical cavities made of two plane-parallel, highly-reflecting surfaces that are slightly spaced -typically from hundreds of Fig. 2 Example of the periodic spectral transmission profile, g, typical of a Fabry-Pérot in a narrow window of ± 1 nm around the central wavelength at which the etalon is tuned for on-axis illumination, λ 0 . Profiles at incidences with respect to the normal of 0°(black), 1°( blue) and 2°(red) are displayed. Note that the resonances are shifted in a non-linear (almost quadratic) way with the incidence angle. Figure  extracted from Bailén et al. (2019a). © AAS. Reproduced with permission micrometers to a few millimeters-. The large reflectivity of the plates causes multiple back and forth reflections across the cavity for each incident ray. Individual rays then split and interfere coherently among themselves. The difference of optical path among two consecutive reflections is such that it causes resonances on the transmission under some conditions. The higher the reflectivity of the surfaces, the larger the electric field amplitude of interfering rays and the narrower the spectral transmission peaks where the constructive interference occur. The conditions of resonance are periodic and depend on the wavelength, refractive index, thickness of the cavity and on the refracted angle of the incident rays (Chabbal 1953). A change on the thickness, index of refraction or the incident direction, can be employed to tune the peaks, although only variations on the two former parameters are usually preferred when the etalon is employed as an spectrometer. In fact, the relative shift of the transmission peak wavelength is simply given by the relative change on the refractive index, n, and/or on the thickness, h. Since the relative change in wavelength necessary to scan a spectral line is typically less than 0.01%, a highly stable control of the thickness and refractive index is crucial for an accurate wavelength sampling. Changes on the direction of the incident light must be minimized too, because the transmission resonances suffer from a displacement to shorter (bluer) wavelengths that increases in a roughly parabolic fashion with the incidence angle (Fig. 2).
The spectral resolution of the Fabry-Pérot also varies with the etalon parameters and with the incidence angle, but in a much lesser extent. However, careful considerations must be taken into account with imperfections on the etalon plates and/or in its cavity (defects). Defects are originated from inhomogeneities in the refractive index (or in the coating) and from variations on the flatness and parallelism of the etalon plates, both of which displace the transmission profile locally. The average result of having individual shifts that vary from point to point across the etalon aperture is a broadening of the transmission profile bandpass that diminishes the resolving power of the filter. The phase of the transmitted electric field is also affected by defects. Errors in the phase reduce the optical quality of the transmitted wavefront. The large number of reflections of the wavefront that take place within the optical cavity of the etalon amplify its degradation. The wavefront error increases rapidly with the reflectivity, which must be limited then in order to achieve diffraction-limited quality (e.g., Kentischer et al. 1998;Scharmer 2006).
Errors are usually divided in two main categories: largescale and small-scale defects. The former tend to increase with the size of the etalon aperture and appear during the manufacturing process or simply because of stress tensions caused by the coatings, by the force of gravity or by the mechanical pressures (Greco et al. 2019). The shape of these errors has been restricted classically as a radial (parabolic) departure of the thickness from the center to the edges -spherical or parabolic defect-or as a linear change in the thickness from one edge to the other along a given direction -parallelism defect- (Reardon and Cavallini 2008). Large-scale defects are more complex in reality and can be better described by fitting a finite set of Zernike polynomials (Greco et al. 2019). As they are influenced by gravity, they also depend on the orientation of the surface of the etalon: vertical or horizontal. Such an orientation will be especially important for the large-format etalons (150 − 250 mm diameter) that are needed in the next generation of solar telescopes (Quintero Noda et al. 2022;Rimmele et al. 2020). The impact of gravity on an aperture as large as 120 mm has been studied recently by Greco et al. (2022). They observed that the contribution of gravity on large-scale defects is dominated by the position of the piezo-electrical actuators for an horizontal mounting, whereas the distribution of defects is much smoother for the vertical orientation. However, the overall magnitude of gravity-induced defects-is similar in the two cases.
Meanwhile, small-scale defects are basically polishing errors of high frequency that are approximately randomly distributed. Figure 3 displays a layout of the geometries for the three mentioned defects. Illumination of the etalon with a non-collimated (i.e., converging or divering) beam also broadens the resonances and shift them towards the blue when compared to collimated illumination (e.g., Sloggett 1984;Atherton et al. 1981). Hence, they are commonly treated as another defect -the aperture defect-.
There are many different approaches to describe the degradation of the spectral resolution caused by defects Fig. 3 Layout of the geometry of the etalon plates when affected by the three most typical defects. From left to right: spherical/parabolic defect with a maximum departure δt s , Gaussian defect with a root-mean-square deviation across the aperture (δt g ) 1/2 , and parallelism defect with an edge-to-edge divergence of δt p (e.g., Chabbal 1953;Meaburn 1976;Hill 1963;Hernandez 1988), but most of them are either limited to the case in which the broadening of the spectral profile is very large compared to the ideal case (e.g., the limiting finesse of Chabbal 1953) or they are only valid to air-gapped etalons (e.g., the aperture finesse of Atherton et al. 1981). The works by Sloggett (1984) and Bailén et al. (2019a) probably provide a more valid description on the influence of the various types of defects on the transmission profile, as their formulation includes the impact of defects of any arbitrary amplitude in both crystalline and air-gapped etalons. The method can be generalized easily to defects with complex geometries, too.

Combinations of etalons
For Fabry-Pérots to work as tunable filters, isolation of only one resonance order of the transmission profile is mandatory. Otherwise unwanted light from the secondary orders can contaminate the signal and degrade the spectral purity of the filter. Narrow-band dichroic (interference) filters with bandpasses comparable to the spectral distance between different peaks of the transmission profile of the etalon (the free spectral range) are commonly employed for that purpose. Usually, they are placed ahead of the etalon, thus receiving the name of pre-filters. The free spectral range (FSR) of the etalon depends quadratically on the wavelength and is inversely proportional to its optical thickness (refractive index times the geometrical thickness), but is usually of the order of several hundreds of pm (e.g., the PHI etalon has a ∼ 300 pm FSR). Commercial off-the-shelf dichroic filters are typically broader. For this reason custom-made solutions are normally needed. Pre-filters with good optical quality, high transmission and with such a narrow bandpass are difficult to manufacture and expensive, though. In addition, they offer a limited out-of-band rejection, since their profile diverges from an ideal rectangular function. Instead, it typically resembles that of an imperfect bell-shaped curve with extended wings.
Such a transmission curve has two consequences on the observations. Firstly, the photon flux is reduced as we move away from the transmission peak, thus degrading the signalto-noise ratio of the observations. A very careful tuning of the pre-filter with respect to the center of the observed spectral line is also needed to avoid the rise of asymmetries in the observed intensity along it. Secondly, the slow decay of the transmission let pass some residual light (∼ 1% or more, typically) from the neighboring resonance orders of the etalon (Fig. 4). This effect is usually refereed as (spectral) stray light or parasitic light and can be minimized by enlarging the FSR or by using two or more etalons with different FSRs. The former is the simplest solution and can be achieved by reducing the cavity spacing, but there are manufacturing, economic and even mechanical constraints that prevent the cavity from being arbitrarily thin. Furthermore, reducing the cavity broadens the bandpass in an inversely proportional way. Another approach consists in using two or three etalons with a careful choice of their cavities to suppress effectively secondary lobes on the transmission profile (Fig. 5).
The first instruments that probably used a system of several etalons successfully were IBIS and TESOS with two and three respectively. 4 The dual-tandem configuration has proven to be the most popular option, though (e.g., IBIS, CRISP, GFPI, and VTF in a future upgrade).
The use of several etalons improves not only the FSR, but also the spectral resolution. In fact, if two identical Fabry-Pérots are placed within the optical path, the resolving power is improved up to a factor √ 2. An alternative arrangement to improve the resolution while keeping the same FSR consists in forcing the optical beam to go through the etalon twice (double-pass configuration). This solution was adopted for IMaX (Álvarez-Herrero et al. 2006), in which the Fabry-Pérot is followed by a system of two folding mirrors that allow light to pass twice through different subapertures (Fig. 6). Although the FSR is unaffected in this configuration, the parasitic light is also reduced by a power of two when compared to a single-pass setup.
Using multiple etalons present an important drawback, though. Since Fabry-Pérots are made of highly reflecting plane-parallel plates, the space between two etalons acts also as a resonant cavity in which multiple internal reflections take place, thus producing a secondary image that is overlapped to the main signal if they are aligned. The spurious image is usually referred as a ghost image and contaminates the measured signal, too. There are several ways to get rid of it: 1. Tuning the cavities of each etalon to reduce to the minimum the amount of ghost signal. This solution has an impact also on the spectral parasitic light. A rigorous discussion on the best choice of the cavity ratios of the two etalons that reduce both ghost and stray-light contamination can be found in Kentischer et al. (1998), Cavallini (2006, and in Scharmer (2006). The optimum ratio have been found to be either ∼ 0.3 or ∼ 0.6, although the precise value depends on each particular instrument. Nevertheless, even if the ghost signals are minimized by an accurate choice of the cavity ratio, the spurious signals can be still prohibitively large to achieve stringent polarimetric sensitivities of 10 −3 (Scharmer 2006).

Placing a low-transmission pre-filter between the etalons.
This is a very effective way to reduce ghost signals farther, since intensity is reduced by a factor T 2 pf , where T pf is the maximum transmission of the filter (e.g., Scharmer 2006). This solution was implemented in IBIS, whose pre-filter reached a maximum transmission of only T pf 30%. Nowadays, interference filters with much higher transmittances (∼ 80%) can be manufactured. Deteriorating intentionally the flux of photons to deal with ghost images increases the observation time needed to accomplish the same signal-to-noise ratio, thus limiting the observation of fast events. For this reason, the prospects of this approach in the next generation of instruments are unclear. 3. Tilting one etalon with respect to the other. This is probably the most effective way of dealing with this problem. It was employed in CRISP and -before its upgrade to a triple-etalon system-in TESOS. 5 This way, instead of reducing the ghost signal, it is moved directly out the instrument detector. Unfortunately, other problems arise when adopting this solution, as it will be explained later.

Two configurations
Historically, there have been two ways to illuminate the etalon in solar instruments: collimated (e.g., Bendlin et al. 1992;Martínez Pillet et al. 2011) and telecentric (e.g. Kentischer et al. 1998;Solanki et al. 2020). The most adequate choice of the configuration has been subject to debate for decades and it continues today, since the performance of each one is inevitably limited in distinct ways. Extensive analyses on the influence on the spectral and imaging properties of both setups can be found in Beckers (1998), Cavallini (1998), Kentischer et al. (1998), von der Lühe and Kentischer (2000), Cavallini (2006), Scharmer (2006)

The collimated configuration
The collimated configuration is characterized for having the Fabry-Pérot placed on a pupil plane, with its image projected to infinity. This way the etalon is illuminated with a parallel beam of rays whose incidence angle varies over the object field (Fig. 7, top). Examples of instruments that opted for this arrangement are IBIS, IMaX, and GFPI. One advantage of this setup is that the illuminated area of the etalon is always the same. In other words, the footprint of the incident beam does not change across the field of view (FoV). Hence, large local errors and aesthetic defects like dust are smoothed out when averaging across the aperture. The optical quality of the interferometer is preserved constant across the image, too.
In return, the different incidence angles on the etalon over the FoV shift the transmission profile peak across the image plane. The main consequence is a field-dependent sampling of the observed spectral line: at the edges of the image the interferometer is tuned to shorter wavelengths of the spectrum than at the center. Individual plate errors and inhomogeneities across the whole clear aperture of the cavity have also an impact on the transmitted wavefront over the whole FoV. And, probably more important, large-scale defects can have a large influence on the performance of the interferometer as their size is comparable to that of the incident beam footprint.

The telecentric configuration
In the telecentric setup, the etalon is located at an intermediate image plane of the instrument while the pupil is imaged into infinity (Fig. 7, bottom). IPM, TESOS, CRISP, PHI and VTF employ this configuration. With this arrangement the chief ray -which by definition passes through the center of the entrance pupil-comes out parallel to the optical axis no matter the incident direction. In consequence, the etalon The finite aperture of the incident cone of rays imposes a limit on the maximum spectral resolution of the transmission profile even if the etalon were free of defects (e.g. Atherton et al. 1981). Illuminating each point of the etalon with rays that come from the pupil with different angles means also that, for a given monochromatic wavelength, the pupil is "seen" by the Fabry-Pérot as if it were inhomogeneously illuminated (Beckers 1998). The effect is commonly referred to as pupil apodization and varies rapidly along the bandpass of the etalon (Fig. 8). The apodization of the pupil changes the monochromatic PSF in an asymmetric way across the transmission profile of the etalon. The asymmetry over the spectral profile brings about undesired signals when measuring the spectrum of the Stokes vector. These spurious artifacts can exceed the polarimetric sensitivity requirements of the instrument (e.g. Beckers 1998;Kentischer et al. 1998;von der Lühe and Kentischer 2000;Scharmer 2006;Bailén et al. 2020). Pupil apodization is very dependent on the f -number of the incident beam and on the resolution of the instrument. The reason is that the thinner the spectral bandpass of the etalon and the larger the incidence angles, the more important is the change of intensity at a monochromatic wavelength across the pupil. Pupil apodization can be minimized, then, by using crystalline etalons -which reduce the refracted angle of the rays-, by sacrificing resolving power or by illuminating the etalon with very large f -numbers. The latter is the solution most commonly adopted. Unfortunately, the size of the instrument and of the etalon itself increase linearly with the f -number of the incident beam. Hence, apertures limited to f/100 − f/200 are normally employed to achieve a compromise between the magnitude of the spurious signals and the size and cost of the instrument (Table 1).

Telecentric vs collimated
As seen above, the two configurations present some benefits and drawbacks. A thorough comparison of the main problems that can arise in the two mounts and how to deal with them is then mandatory to decide whether we should choose one setup or the other.

Limitations of the telecentric setups
Pupil apodization The first work to address consequences of the spectral dependence of the PSF across the bandpass of the etalon characteristic of telecentric etalons was the one by Beckers (1998). He showed that the PSF width decreases at wavelengths towards the blue of the transmission peak at the expense of transferring the energy to its wings, whereas towards the red part of the passband the opposite is true: the core of the PSF gets broader while the lobes are reduced to conserve the total energy enclosed by the PSF (Fig. 9). This effect increases in a nonlinear way with the resolving power of the spectrometer. According to his numerical estimations, the contamination introduced by such an asymmetric spectral response could be as high as 30 ms −1 in regions with no real Doppler shifts. Such signals are impossible to correct fully with post-processing techniques of the measured data and led him to conclude that collimated setups should be used in magnetographs that aim to the highest performance. Bailén et al. (2020) demonstrated later that spurious signals on the inferred magnetic field are also expected to arise in these mounts. The magnitude of these scale approximately also as ∼ (f #) −2 , where f # is the f -number of the incident beam.
Phase errors Although the quantitative predictions of Beckers (1998) on the spectral behavior of the PSF in a telecentric mode were correct, he did not consider fluctuations on the phase of the transmitted electric field (phase errors) originated by pupil apodization, but only the changes in the module of the electromagnetic field. Phase errors were included later by von der Lühe and Kentischer (2000), who showed that they foster a transfer of energy between the central part of the PSF to its lobes. They also degrade the wavefront, thus limiting the imaging performance of this configuration even if no defects were present on the etalon. Phase fluctuations evaluated at the maximum of the transmission profile exhibit a parabolic trend with the radial coordinate of the pupil. Therefore, Scharmer (2006) proposed a simple refocusing of the final image plane to correct them to a large extent. Bailén et al. (2021) tackled the impact of phase errors including the full polychromatic response of the etalon. Fig. 9 Cross section of the PSF (in logarithmic scale) along the transmission profile of an f/150 telecentric etalon with a ∼ 4 pm spectral resolution. The Y axis represents the spatial dimension of the PSF, in units of the Airy disk radius (ρ 0 = 1.22λ/D). The X axis accounts for the spectral dimension, centered about the peak of the transmission profile, λ p They found that the wavefront degradation is expected to be less dramatic than the one presented by von der Lühe and Kentischer (2000) and Scharmer (2006) after integrating along the whole bandpass and can be partly overcome with the refocusing proposed by the latter. Pupil apodization effects cannot be mitigated this way, though.
Etalon defects Local errors on the thickness and/or the cavity homogeneity with sizes comparable to the footprint of the incident beam invalidate the use of a PSF from a strict point of view, since the response of the etalon is expected be field dependent (Bailén et al. 2019a). The same occurs with the spectral transmission, which suffers from local shifts across the clear aperture. The telecentric mount thus loses one of its main advantages over the collimated setup: the preservation of the same spectral response across the FoV. The field-dependent variations of the PSF and bandpass of the etalon can be smoothed out by defocusing the Fabry-Pérot, which necessarily increases the footprint of the incident beams and averages local errors over a larger area. In return, the impact of defects on the wavefront will increase.

Telecentrism imperfections
Errors on the telecentrism produce a relative tilt between the incident chief ray and the normal of the etalon plates that varies over the FoV. The imperfection can arise because of departures from first-order optical properties (paraxiality) or from deviations of the optical elements from their nominal position, both occurring in real instruments. The off-axis incidence of the chief ray on the etalon produces a loss of the axial symmetry on the pupil apodization and induces field-dependent asymmetries, broadenings and shifts on both the PSF and the transmission profile (Bailén et al. 2019a). The imaging and spectral degradation introduces again systematic errors in the inferred magnetic field and in LOS plasma velocities. The largest spurious signals can be as high as ∼ 5% for imperfections of only a few tenths of degree and for f -numbers and spectral resolutions typical of solar instruments (Bailén et al. 2020). 6 Tilts of the etalon Deviations on the incidence of the chief ray from normal illumination of the etalon plates also occur when tilting the Fabry-Pérot to move ghost images out of the detector, with an impact on the PSF and transmission profile equivalent to the one described in the previous paragraph, but constant across the FoV in this case. Since pupil apodization effects are strengthened with increasing resolving powers, the tilt should be kept as close as possible to the limiting value that ensures that the ghost image is moved away from the detector. Moreover, in dual-etalon instruments the Fabry-Pérots are preferred to have different spectral resolutions to apply the tilt to the one with lowest resolution (Scharmer 2006). Yet, asymmetries in the PSF and in the spectral profile cannot be eliminated completely as long as one etalon is inclined.

Mutual masking
In systems with multiple etalons, the individual transmission profiles of each Fabry-Pérot are shifted differently from one point to another because of the presence of local differences in their individual cavity maps. This effect is known as mutual masking (Mack et al. 1963). In telecentric setups, these differential detunings bring about locally-changing asymmetries on the final spectral profile, as well as a point-to-point decrease of the transmission (Fig. 10). Variations of the transmission across the aperture can be corrected through flat-fielding procedures during the data processing pipeline, but the signal-to-noise ratio of the observations will be still compromised in a different manner from one point of the FoV to another. The calibration of point-to-point asymmetries is also possible to some extent, although it requires more sophisticated reduction methods (e.g., de la Cruz Rodríguez et al. 2015).

Limitations of collimated setups
Amplification of errors Wavefront errors caused by plate defects are expected to be amplified more than in the telecentric case, in general. The reason for this is that the footprint of the incident beam is much larger and encompasses not only small-scale defects, but also large-scale errors, whose magnitude might be comparable or even larger (e.g. Schmidt et al. 2016). A quantitative assessment on this topic was studied first by Ramsay (1969) and then by von der Lühe and Kentischer (2000), both considering strictly monochromatic Fig. 10 Individual transmission profiles for a high-resolution etalon with a reflectivity of 94% (orange solid line) and for a low-resolution etalon with a reflectivity of 86% (green solid line) in a telecentric f/150 configuration. The final profile obtained from the product of the two transmissions is also displayed (blue solid line). We have introduced a small error on the optical thickness of the low-resolution etalon (∼ 10 −7 %) to simulate the local differential variations on the cavity map expected in real multiple-etalon systems. The detuning produces a significant decrease of the transmission, as well as discernible asymmetries on the final profile incident collimated beams. Later, Scharmer (2006) included the effects of integrating over the whole spectral bandpass of the etalon and found that the polychromatic nature of the observations relaxes the amplification of errors approximately to a half. Still, the results of von der Lühe and Kentischer (2000) and Scharmer (2006) are pessimistic about the use of a collimated configuration and suggest the use of the telecentric solution to achieve diffraction-limited performance, contrary to Beckers (1998).

Field-dependent spectral shift
The response varies across the FoV because of the different incidence angles on the etalon, like in the (real) telecentric configuration, although in this case the spatial shape of the monochromatic PSF is preserved and the variation of incidence angles manifests itself only as a continuous radial shift of the transmission profile. This displacement can be calibrated easily as long as it is kept smaller or comparable to the spectral sampling interval of the instrument (e.g., Cavallini 2006).

Mutual masking
In systems with several etalons, the local detuning of the individual transmission profiles described earlier has also an influence on the collimated mount. Firstly, differential shifts of the spectral profiles produce a decrease and a broadening of the overall transmission of the instrument. Secondly, the presence of asymmetric defects of large scale -due, for instance, to local stresses caused by the piezo electric actuators used to control the etalon thickness-is responsible for an asymmetrization of the final transmission profile, since it is no longer given by the product of the individual spectral profiles, but by the average of the product of the local individual transmissions (Reardon and Cavallini 2008). These two effects are expected to have a larger impact when increasing the aperture of the etalon.

Field-dependent detuning
In systems with multiple etalons, a field-dependent differential shift among the individual transmissions appears when one of them is tilted to suppress ghost images (Cavallini 2006;Scharmer 2006). The relative spectral detuning arises from the difference in incidence angles on each etalon and produces a field-dependent asymmetry on the transmission profile and a shift similar to the one occurring in a tandem of telecentric etalons because of local changes on the optical cavity map. The difference being that in this case the dependence can be modeled in a known way over the FoV.
Instead of tilting one etalon, the aforementioned solution of placing a pre-filter with low transmission among etalons proposed by Cavallini (2006) can be applied in this case to reduce the inner-etalon ghost signal intensity without detuning the etalons. However, apart from deteriorating the flux of photons, introducing the pre-filter between the two etalons has an additional problem: it must be of the same size of the etalons ( 50 mm). Such large interference filters are difficult to manufacture with reasonable optical qualities of ∼ λ/20 or better to achieve diffraction-limited performance of the instrument. The pre-filter profile would also shift differently across the FoV with respect to those of the etalons because of the unavoidable difference in the indices of refraction, which causes a distinct response with the incidence angle. The displacement on the transmission profile induces, again, field-dependent asymmetries and shifts of the total transmission. To minimize the relative variation of the profiles over the FoV, a pupil adapter (e.g., Greco and Cavallini 2013) should be employed to reimage the pupil on the prefilter with different size, thus changing the maximum incidence on the pre-filter according to the Lagrange invariant to produce a similar wavelength shift. The trade-off for using an adapter is that it adds more optical elements, with the consequent impact on the optical quality and, especially, on the size of the instrument.

Discussion
In collimated instruments, cavity errors and inhomogeneities are smoothed out across the clear aperture of the etalon and the artifact arising from the spectral dependency of the PSF do not appear at all. Unfortunately, this mount is not exempt of problems either. Among the mentioned drawbacks of the collimated mode, the most critical to tip the scales in favor However, the predictions of Scharmer (2006) about the collimated setup seem too hopeless to us, since his conclusions are based on considering highly-reflective etalons that suffer from (unavoidable) microroughness thickness deviations with a root mean square (RMS) value of 2 nm, while etalons with errors in the range of 0.5 − 1 nm (including large-scale defects) are feasible to manufacture nowadays, at least for etalons of modest sizes of the order of 50 mm (e.g., Álvarez-Herrero et al. 2006;Greco et al. 2019). Furthermore, part of the wavefront deformation can be compensated for by refocusing the instrument, especially the defocus term introduced by the parabolic defect contribution. Departures from parallelism can be addressed, too. In fact, an effective method to correct dynamically the loss of parallelism in servo-controlled etalons has been presented recently by Greco et al. (2019). Figure 11 shows an example of the cavity error map measured by Greco et al. (2019) after applying their tilt correction technique to a real 50 mm aperture etalon. The residual large-scale and small-scale contributions are only ∼ 0.6 nm and ∼ 0.7 nm, respectively, summing a total of ∼ 0.9 nm RMS error.
For large-format etalons, the prospects are very promising, too. Greco et al. (2022) have recently observed that the contribution of gravity in defects of large scale can be as low as 0.3 nm RMS for a "heavy" commercial etalon of 120 mm clear aperture diameter when mounted either vertically or horizontally. Small scale errors have a contribution of only ∼ 0.5 nm RMS across the whole clear aperture.
Excluding large scale defects, residual errors of the order of ∼ 0.5 nm or so -four times smaller than the ones modeled by Scharmer (2006)-seem, then, feasible to achieve (e.g., Reardon and Cavallini 2008;Greco et al. 2019Greco et al. , 2022. Novel manufacturing techniques can minimize microroughness defects even more, although at the expense of increasing dramatically their cost. Such is the case of the large VTF etalon, whose microroughness plate errors are smaller than 0.4 nm RMS across its large 250 mm aperture (Sigwarth et al. 2016). Of course, other more complex largescale variations of the geometry apart from the parabolic and parallelism defect are usually present and they can be corrected only with image reconstruction techniques (Reardon and Cavallini 2008;Greco et al. 2019). Sometimes their magnitude is similar or larger than the one of microroughness errors, which could prevent the optical performance of the etalon to reach the diffraction limit before post-facto image reconstruction. However, we believe that, in general, it should not be stated that collimated etalons are not suited for diffraction-limit applications. Instead, it is crucial to explore whether tight requirements on the small-scale and the contributions of large-scale defects that are impossible to correct can be accomplished by the manufacturer or not and to which extent they can be corrected for with image reconstruction methods. It is also mandatory to characterize in detail the map of cavity defects once supplied by the manufacturer, since cavity geometries can vary substantially even among similar etalons produced by the same company (Greco et al. 2019).
A similar argument applies for telecentric setups. Their most important disadvantage is probably the impact of pupil apodization effects, but the conclusions drawn by Beckers (1998) are too categorical from our point of view. The magnitude of the artificial signals induced by these setups depends greatly on the product of the f -number of the incident beam times the refractive index of the etalon, as well as on the spectral resolution of the spectrometer (Bailén et al. 2021). Whether the use of a beam slow enough to keep the signals below the sensitivity requirements of the instrument is compatible with the available dimensions or not needs to be assessed in each case. The influence of asymmetries and shifts of the PSF and the transmission profiles should be evaluated in detail, too.

General overview
Typically, etalons are controlled by piezo-electric (servocontrolled) actuators that accurately adjust the thickness of the cavity for tuning purposes (e.g., Kentischer et al. 1998;Puschmann et al. 2006;Scharmer et al. 2008). However, there is an alternative form to change the cavity of the Fabry-Pérot: crystalline etalons can be tuned by applying an electric field only. These etalons are filled with a crystal that lacks of a center of symmetry (noncentrosymmetric). They therefore exhibit linear (Pockels) electro-optical and piezoelectric properties that allow for a change of both the refractive index and the cavity thickness in the presence of a differential voltage between the cavity plates. The main advantages of crystalline etalons are that: 1. they prevent from vibrations of the etalon caused by mechanical (piezo-electric) actuators, 2. their weight is low, and 3. their refractive index is usually much larger than that of air.
These features have direct benefits in space applications, the two first being obvious. Regarding the third one, large refractive indices reduce the refracted angle within the cavity, which alleviates the field-dependent shift of the transmission characteristic of the collimated configuration, as well as pupil apodization effects inherent to telecentric setups (e.g. Debi Prasad and Gosain 2002). The incidence angle and the f -number on the etalon are inversely related to its diameter in the two configurations through the Lagrange invariant (Cavallini 2006). This means that crystalline etalons need a much smaller aperture to achieve a similar performance than air-gapped Fabry-Pérots. The use of larger incidence angles typically allows to down-scale the dimensions of the entire instrument. Such an advantage is particularly helpful in space, where tight requirements on the dimensions of the instrument are applied. The use of solid etalons could also be an asset in instruments for the large-aperture ground-based telescopes, where the large clear apertures of the order of 150 − 250 mm expected if filled with air (Schmidt et al. 2016;Greco et al. 2022) can be reduced by a factor of two or more. Currently, LiNbO 3 is the material par excellence in space instrumentation (Martínez Pillet et al. 2011;Solanki et al. 2020). 7 Other crystals, such as MgF 2 have been employed in space etalons too, but only combined with piezo-electric actuators (Gary et al. 2006). The use of the LiNbO 3 technology in both space applications and in the next generation of ground based telescopes can decrease the etalon dimensions -and the length of the whole instrument-by a factor ∼ 2.3, with the benefits mentioned above.
Unfortunately, solid etalons present also several limitations. The most remarkable are probably the following: 1. The large refractive index decreases the free spectral range and amplifies wavefront degradation compared to an air-spaced etalon of equal thickness. The cavity of the Fabry-Pérot must be shortened to overcome these issues, but there are physical limitations that prevent from the use of arbitrary small cavities. For example, the manufacturing of LiNbO 3 crystalline etalons with thicknesses 7 Michelson interferometers have also been successfully employed in space-borne applications (Scherrer et al. 1995(Scherrer et al. , 2012, but they present other drawbacks mentioned in Sect. 1. below ∼ 200 μm has proven to be challenging in terms of mechanical stability (Álvarez-Herrero et al. 2006).

The tuning speed of the etalons is slow if compared
to that of piezo stabilized Fabry-Pérots. In particular, LiNbO 3 etalons reach maximum tuning speeds of the order of 50 pm s −1 (Martínez Pillet et al. 2011), whereas servo-controlled Fabry-Pérots are about one to three orders of magnitude faster (e.g., Greco et al. 2022). 3. They need very high voltages -of the order of a few thousand volts-to scan a whole spectral line. 8 The tuning range can be increased by decreasing the thickness of the etalon (Eq. (28) of Bailén et al. 2019a) but, again, only to a certain extent. The usual limit of the tunable spectral range is roughly to ±0.1 nm. Meanwhile, airgapped etalons are only limited by the maximum translation achievable by the piezo-electric actuators and can cover tuning ranges several times broader (e.g., Greco et al. 2019). 4. They are very sensitive to small variations of the temperature, so they must be controlled thermally. 9 5. When used in balloon-borne instruments they shall be pressurized, in order to avoid electrical arc discharges originated by the dielectric breakdown of the surrounding air, occurring at millibar pressures (Martínez Pillet et al. 2011). 6. The majority of noncentrosymmetric crystals are anisotropic. In fact, crystals belonging to the trigonal, tetragonal and hexagonal systems (e.g., LiNbO 3 ) are uniaxial -i.e., they have a preferred direction along which electromagnetic waves propagate as if they were in an ordinary isotropic medium-, whereas those categorized within the orthorombic, monoclinic or triclinic system are biaxial. As anisotropic materials, they can change the polarization state of light. In other words, they are birefringent. 10

Anisotropy as a source of artificial signals
Crystals with a unique optical axis oriented orthogonally with respect to the etalon plates are preferred, since birefringent effects can be minimized simply by illuminating them with beams as close to the normal of the reflecting plates as possible (Martínez Pillet et al. 2011;Solanki et al. 2020). Etalons with such an orientation of the optical axis are said to have a Z-cut configuration, as opposed to the Y -cut arrangement, in which the optical axis is parallel to the plates (e.g., Netterfield et al. 1997). The Y -cut configuration has been employed in solar applications as well (Feller et al. 2006;Kleint et al. 2011), but it has demonstrated to be practical only if the etalon is placed after the linear polarizer of the polarimeter, with the consequent disadvantages that will be commented below. Even when the Z-cut configuration is chosen to avoid birefringent effects, etalons are illuminated by either converging (telecentric) light or with collimated light with incidence angles that vary over the FoV. Cross-talks among different Stokes parameters that can corrupt the measurement of the polarization state of light are, then, expected (Fig. 12). When considering local defects of the etalon, such a contamination is field-dependent even in an ideal telecentric setup. Birefringent effects can also appear even in normal illumination conditions due to the existence of local regions in which the optical axis is misaligned with respect to the Z direction (local domains). Such deviations may appear during the manufacturing or from hysteresis processes that can take place when fed with very high voltages. The latter effect has been observed in the spare LiNbO 3 etalons of the PHI instrument when voltages of ∼ 4 kV or more are applied.
An experimental evaluation of the influence of the birefringent properties of a uniaxial (liquid crystal) Fabry-Pérots on the transmission profile can be found in Vogel and Berroth (2003), while numerical algorithms that model the propagation of electric fields in crystalline etalons is performed in Zhang et al. (2017). 11 Meanwhile, studies on the artificial signals that take place because of the polarimetric properties of the multilayer plate coatings has been carried out numerically by Doerr et al. (2008) and Ejlli et al. (2018). According to Doerr et al. (2008), the largest deviation on the PSF introduced by the birefringence of the coatings is only of the order of 10 −6 for etalons illuminated with typical incidence angles characteristic of solar instruments.
The influence of crystalline etalons in spectropolarimeters was studied for the first time probably by Del Toro Iniesta and Martínez Pillet (2012), who considered the effect of solid etalons by postulating that their Mueller matrix is simply given by the sum of a retarder and a mirror. With this premise, they demonstrated analytically that a spectropolarimeter formed by a set of two nematic LCVRs as modulator, a birefringent Fabry-Pérot as tunable filter and a linear polarizer as analyzer can still reach the maximum polarimetric efficiencies with proper modulations of the pair of LCVRs. Formal derivations of the Mueller matrix of crystalline etalons that take into account the full nature of the problem have been presented more recently by Bailén et al. (2019b). Bailén et al. (2019b) evaluated the impact of the anisotropy in each optical configuration: collimated or telecentric. When the collimated configuration is chosen, the direction of the incident beam varies for each point of the observed object field and, with it, the birefringence induced in the etalon. For this reason, its Mueller matrix is expected to change from point to point across the field. In an ideal telecentric configuration the response of the etalon would be constant along the FoV, but imperfections on the illumination and defects on the crystal or in the plates of the etalon cause a field-dependent polarimetric response of the etalon if this configuration is used, much like in the collimated case. Moreover, deviations from perfect telecentrism, or tilts of the etalon to deal with ghost images, are responsible for an asymmetric spectral performance of the Mueller matrix, similar to the one appearing in the transmission profile of isotropic etalons.
Fortunately, the etalon is always placed either between the modulator and the analyzer of the polarimeter (e.g., IMaX) to allow for the use of dual-beam methods with a single etalon, or right after the polarimeter (e.g., PHI). Modulation of the incident illumination on the etalon and the presence of a linear polarizer right after the etalon are expected to reduce birefringent effects to a large extent (Bailén et al. 2020). In the collimated configuration, spurious signals arising from the anisotropy of the crystal can be eliminated completely when the etalon is situated after the polarimeter at the cost of disabling dual-beam polarimetry, unless two identical etalons are used, one for each of the channels of with it their birefringent properties. They also suffer from a non negligible absorptivity that reduces considerably the maximum achievable transmission. the instrument. However, this solution increases the cost and weight and makes more difficult to calibrate, operate and interpret the data retrieved by the instrument. In the telecentric configuration the artifacts cannot be removed by placing the etalon after the polarimeter as its Mueller matrix commutes with that of a linear polarizer, but the birefringent contribution is expected to be much less important than the one associated to pupil apodization effects (Bailén et al. 2020). In any case, the polarimetric response of the etalon should be taken into account combined with that of the rest of the instrument in order to minimize its influence or, at least, to calibrate it.

Conclusions
We have reviewed the performance of Fabry-Pérot etalons taking into account their spectral, imaging and polarimetric properties. We have discussed about the use of single and multiple systems etalons in solar instrumentation. We have paid special attention to the way they are illuminated -collimated and telecentric-, and we have highlighted the benefits and drawbacks of each configuration. Finally, we have compared the performance of crystalline and airgapped etalons, focusing on the birefringent effects introduced by the former technology.
Instruments equipped with a single etalon benefit from a simpler design and an easier operation. They are also lighter and less power-consuming. However, they need to be combined with very narrow pre-filters that are difficult to manufacture and, hence, expensive. These pre-filter can introduce an important amount of residual parasitic (or stray) light coming from secondary orders. The bell-like transmission curve of the pre-filters can reduce also the signal-to-noise ratio of the observations at different tuning wavelengths. Systems of two or three etalons reduce effectively the parasitic light and improve the spectral resolution of the instrument, but they can produce ghost images on the detector. There are different ways to overcome these artificial images, the most effective being a tilt in one of the etalons to move them away from the detector.
No matter the number of etalons, they are usually arranged in two different configurations: collimated and telecentric. In the former the etalons are placed in one image of the pupil of the instrument, in collimated space. In telecentric setups, etalons are located in an intermediate image plane while the pupil is projected to infinity.
The main advantage of the collimated configuration is that it offers both the same spectral and imaging response over the field of view, except for a shift. Unfortunately, it presents two major limitations: (1) large-scale defects on the etalons have a large influence on the optical quality; and (2), a field-dependent detuning that induces asymmetries in the spectral and imaging profiles arises in systems of two etalons if tilted to avoid the emergence of ghost images on the detector. The first problem can be overcome by tightening the requirements on the optical quality of the etalons. The second one can be minimized only by reducing the amount of tilt in one etalon.
The telecentric configuration is safer in terms of wavefront degradation, but it suffers other downsides. The most important is probably that pupil as apodized at monochromatic wavelengths in a different manner along the transmission profile. This originates a spectral dependence of the PSF responsible for spurious signals on the measured magnetic field and plasma velocities and introduces phase errors. The latter can be dealt to some extent by refocusing the etalon, but the artificial signals are much more difficult to correct and increase rapidly with the inverse of the fnumber and with the tilt of the etalon. This mount introduces also a field-dependent response caused by small-scale errors on the etalon, by imperfections on the telecentrism, or simply because a tilt is applied to one of the etalons to deal with ghost images.
The use of one configuration over the other is somehow a matter of personal taste, since it depends on the preferences of the designer. If the wavefront quality is to be pushed to its limits, the telecentric configuration is usually preferred. If an homogeneous response over the field of view is prioritized and artifacts on the observed signals need to be minimal, collimated mounts are better suited. Limitations in the capabilities of the etalon manufacturer, on the available financial budget and on the overall dimensions and weight of the instrument play an important role in the decision, too.
Two technologies are employed in current instrumentation: air-gapped and crystalline etalons. Although most of the etalons employed in solar instruments are filled with air, LiNbO 3 etalons are the standard in space applications. Their popularity comes from their low weight, the elimination of vibrations associated to mechanical actuators, and their large refractive index. The latter enables their use with larger incidence angles, which means also that, when compared to air-gapped etalons, (1) they need smaller clear apertures and (2) they shorten the overall size of the instrument. This technology is not exempt of issues, though. Firstly, they are several orders of magnitude slower than piezo-electric etalons. Secondly, their thickness needs to be of only a few hundreds of microns to produce a large enough free spectral range and to keep the impact of defects degradation under control. And, thirdly, they modify the spectropolarimetric properties of light, thus introducing spurious signals on the physical quantities inferred on the instrument. However, the latter are expected to be small enough when taking into account the complete modulation scheme of the instrument and can be reduced to the minimum if the etalon is placed after the analyzer.
Author contributions F.J. Bailén wrote the main manuscript text. D. Orozco and J.C. del Toro reviewed the manuscript.
Data Availability Not applicable.

Materials Availability Not applicable.
Code Availability Not applicable.

Declarations
Ethics approval Not applicable.

Competing interests The authors declare no competing interests.
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