Throughout the paper, we use spectroscopic terminology that might have slightly varying meanings in different fields of spectroscopy. To avoid confusion, the terms are briefly explained here.

A spectrograph is a spectrometer where the components of the spectrum are separated in space and recorded simultaneously with a detector array. The instrument line function (ILF) H describes the response of a spectrograph to an input of spectrally infinitesimal width (i.e. monochromatic radiation). The ILF determines the spectral interval that can be resolved by the spectrograph. In the following, this interval is called a spectral channel of the spectrograph (not to be confused with the spectral range covered by a pixel of the spectrograph's detector). Its full width at half maximum (FWHM, denoted by δλ) can be used (amongst other and rather similar definitions) to quantify the spectral resolution. What we call high spectral resolution corresponds to a narrow width of a spectral channel (i.e. a low value of δλ). The spectral range covered by all spectral channels of the spectrograph describes its spectral coverage. The resolving power R of the spectrograph is the ratio of the operating wavelength λ to the spectral resolution δλ. Investigating the light throughput of spectrographs on a spectral channel basis allows the direct comparison of their noise-limited detection limits for trace gas absorption (see Sect. ).

In spectroscopic atmospheric trace gas remote sensing, the column density S of the gas is directly quantified. The column density denotes the concentration of the trace gas integrated along the respective measurement light path. According to different experiment designs and applications, the light path differs and ultimately determines the detection limit in terms of concentration see e.g.for details.

The width of rovibronic absorption lines of atmospheric trace gas molecules in the near-UV to NIR spectral range as well as that of many Fraunhofer lines are of the order of some picometres. In order to observe the corresponding spectral structures (in particular individual rotational lines), resolving powers in the range of R≈105 are required. This defines what we refer to in the following as “high spectral resolution”.

In the UV and visible spectral range many trace gas molecules show “bands” of absorption lines composed of many, partially overlapping rotational lines of a vibrational transition, resulting in structured absorption cross sections, even when observed with moderate spectral resolution (R≈1000). These trace gas molecules can be quantified along light paths inside Earth's atmosphere by differential optical absorption spectroscopy DOAS; see. Compact moderate-spectral-resolution GSs are used to record spectra of direct or scattered sunlight or artificial light sources from ground-based to space-borne platforms and, thus, allow for spatially and temporally resolved measurements of (also very reactive) trace gases.

However, a higher spectral resolution is desirable in many cases. There are atmospheric trace gases that are more difficult or even impossible to measure with moderate resolution. For instance, hydroxyl radicals (OH) exhibit distinct and narrow absorption lines (widths of 1–2 pm at 308 nm; see Fig. ). Due to the low atmospheric concentrations of this species, its absorption can not be separated from overlaying effects (e.g. other absorbing gases) with spectral resolutions that are much lower than the width of the individual lines. Tropospheric OH concentrations have been measured with high-resolution absorption spectroscopy by studies such as , , and using large GS set-ups (850–1500 mm focal length) and an intricate broadband laser system as a light source as described in. Direct sunlight measurements of OH have been performed with Fourier transform spectrometers FTSs; e.g., a high-resolution GS 1500 mm focal length;, and rather delicate systems employing series of pressure-tuned FPIs e.g.. Furthermore, high-resolution O2 measurements have been performed in the atmosphere (e.g. ) using a GS (1500 mm focal length). The high spectral resolution allows one to quantify the absorption of individual lines of different strength and, therefore, to infer, for instance, the light path length distributions in clouds. The rather complex and immobile hardware of the named measurements limited their application to a few and locally restricted atmospheric studies.

Many other atmospheric trace gases show strong and structured absorption on the picometre scale. Besides sulfur dioxide SO2; e.g., formaldehyde HCHO; e.g., water , and chlorine monoxide (ClO; ), found strong, discrete, and narrow bromine monoxide (BrO) absorption lines in the UV region. Using these much more detailed and specific spectral features of the trace gases could not only substantially increase the selectivity but also, in many cases, increase the sensitivity of DOAS measurements. Additionally, the absorption cross sections of isotopologues of some trace gases could be distinguished, similarly to the moderate-spectral-resolution measurements of water vapour isotopologues e.g.. Figure a illustrates the addressed difference in spectral resolution by showing the high-resolution absorption cross section of SO2 as well as a convolution representing the absorption cross section as seen by a compact GS with a 0.4 nm spectral resolution.

Moderate-resolution scattered sunlight DOAS measurements largely undersample solar Fraunhofer lines (the width of which can also be in the picometre range). On the one hand, this introduces uncertainties in the effective spectral absorption of the trace gases see e.g.; on the other hand, in most cases, it implies the need for a Fraunhofer reference spectrum. High-resolution spectra would allow a direct separation of Fraunhofer structures from narrow trace gas absorption structures; moreover, absolute atmospheric column densities of trace gases could be determined (rather than the column density relative to a reference spectrum).

A key point in the success of moderate-spectral-resolution DOAS measurements in the atmosphere is the use of compact and stable (i.e. mobile) spectrographs (volume of the order of 1 L, a focal length f of about 10 cm, and no moving parts). As mentioned above, they typically yield a resolving power of approximately 1000 and a light throughput that allows for the recording of scattered sunlight spectra in the UV and visible spectral range with a SNR of several thousand within less than a minute e.g.. This is sufficient to retrieve many of the weakly absorbing atmospheric trace gases in the UV and visible spectral range (optical densities of ca. 0.01–0.0001) and to study their dynamics and chemistry.

The mobility of measurement equipment provides substantial advantages for practical field applications, including the following: (1) deployment on mobile platforms (e.g. cars, camels, drones, balloons, aircraft, and miniature satellites); (2) the significant reduction of costs for field campaigns due to reduced infrastructure and human resource requirements; (3) remote locations (e.g. deserts or volcanic craters) are made accessible (e.g. with backpack sized instruments); and (4) instruments can be employed in autonomous, remote, and low-maintenance measurement networks see e.g.. In practice, these points are substantial factors making scientific environmental observations feasible.

As will be shown below, increasing the resolving power of a GS also requires a larger instrument size. Thus, the mobility advantages are largely lost. The use of FPIs in spectrograph set-ups can yield high resolving power while maintaining a high instrument mobility.

This work focusses on spectrograph set-ups (GSs, FPI spectrographs) because of their high stability (no movable parts) and low sensitivity to fluctuations in light intensity. FTSs (i.e. Michelson interferometers) do not fulfil these requirements. A one-pixel detector records interferograms in a temporal sequence while mechanical changes in the optics (i.e. the interferometer path length) are conducted. This already imposes limitations on the mobility of the FTS as well as its applicability under more dynamical measurement conditions (e.g. cloudy skies). As (in addition to GSs) FTSs are in broader use in atmospheric remote sensing (mostly towards longer wavelengths, where the well-established and cost effective technology of silicon detector arrays can not be used anymore, i.e. above ca. 1100 nm), they shall nevertheless be briefly mentioned here.

In contrast to GSs and FPI spectrographs FTSs reach a large spectral coverage with very high and adjustable spectral resolution. This can be an important advantage for many atmospheric studies.

compared the SNR of high-resolution (R≈300000) FTS measurements to the SNR of GS measurements with a similar resolving power for direct sunlight measurements at around 308 nm. It was found that the SNRs of the FTS and GS were similar for clear-sky conditions and worse for the FTS under hazy or slightly cloudy conditions. Thus, the advantages of FPI spectrographs regarding the SNR found below (see Sect. ) are expected to similarly hold for an FPI spectrograph to FTS comparison. While the spectral coverage for the high resolution of FTSs is superior, mobility aspects (movable parts and large focal lengths in FTSs) clearly favour FPI spectrographs.

A simple and compact FPI spectrograph can be implemented with a static FPI as well as optics that image the different FPI incidence angles of the traversing light beam to concentric rings of equal FPI transmission on the focal plane (see Fig. a). There, a detector array records the intensities of the different spectral channels simultaneously. The spectral shift of the FPI transmission due to a small change in the small incidence angle α (i.e. a few hundredths of a radian, α≈sin⁡α≈tan⁡α, cos⁡α≈1) is dependent on the wavelength λm of the transmission peak of the order m and α itself (see Eqs.  and ): 4dλmdα=2dnmddαcos⁡α=-2dnmsin⁡α≈-λmα. This demonstrates the non-linearity of the dispersion, which, however, leads to a constant light throughput for all spectral channels (as described in detail below). The wavelength range Λm covered by a particular transmission order (i.e. the FPI's spectral tuning range) is determined by the angle range covered by the parallelised light beam traversing the FPI: 5Λm=-λm∫αminαmaxdαα. The maximum and minimum incidence angles, αmax and αmin respectively, are determined by the illuminated entrance aperture B and the focal length of the collimating lens of the FPI spectrograph's imaging optics (lens 1; see Fig. 2a). For the imaging axis centred at the optical axis (αmin=0), the maximum incidence angle is 6αmax≈B2f1. Assuming, for instance, an entrance aperture of B=3mm and a focal length f1=50mm, the maximum FPI incidence angle would be αmax=0.03 (or 1.72∘) and the spectral coverage would be about 0.135 nm at 300 nm. The practical incidence angle range that can be imaged onto the focal plane is in the range of a few degrees; therefore, the wavelength coverage can typically reach some hundreds of picometres in the near-UV. A moderate-resolution FPI spectrograph (with R≈1000, i.e. a spectral resolution of some hundreds of picometres at 300 nm) of the proposed implementation would exhibit a spectral coverage of the order of its spectral resolution, which would render it rather useless. This problem could be solved by tilting the FPI with respect to the imaging optical axis. Moderate-resolution FPI spectrographs are, however, not addressed in this study. For a resolving power of about 105, the 0.135 nm wavelength range at 300 nm would be divided into about 45 spectral channels with a 3 pm spectral resolution. This is about the number of spectral channels used in a typical moderate-resolution DOAS fitting window.

The sampling of the different spectral channels can be adjusted via the detector pixel size and the focal length of lens 2. Due to the non-linear dispersion, the sampling needs to be adjusted to the outermost ring corresponding to the spectral channel with the lowest wavelength of an FPI order (when assuming equally sized pixels). For the above example (B=3mm, f1=50mm) and f2=50mm, the radial extension of the outermost spectral channel is about δλFPIλαf2≈17µm (see Eq. ). Nowadays, detector pixels with a 1–5 µm pitch are common. This would facilitate sufficient sampling (>3.4 pixels per spectral channel width) for all spectral channels. The spectral sampling can further be adjusted via the focal length of lens 2. As the intensities of all pixels with the same wavelength are co-added, this does not affect the light throughput.

The above-mentioned order-sorting mechanisms (OSMs) allow two basic FPI spectrograph implementations:

Using a grating as the OSM in an FPI spectrograph results in a superposition of the linear grating dispersion with the radially symmetrical FPI transmission on the detector (see Fig. c). This allows one to record several FPI transmission orders at once, thereby increasing the total spectral coverage of the FPI spectrograph. This OSM is referred to as a grating OSM in the following.

An optical fibre and, as the case requires, relay optics direct the light collected by a telescope to the entrance aperture B (see Fig. a). From there, it traverses the imaging optics, containing the FPI and the OSM (certainly, the OSM can also be in front or behind the FPI imaging optics, for instance, the focal plane of an order-sorting GS could be re-imaged).

Both OSM implementations allow for simple, stable, and mobile set-ups with no moving parts. Therefore, they can be applied similarly to moderate-resolution compact grating spectrographs in field measurement campaigns, autonomous measurement networks in remote areas, and in airborne or satellite applications.

When examining spectroscopic methods, a basic question is how a physical parameter changes as a function of the wavelength λ. For spectrographs, this physical parameter is most often a deflection angle θ(λ) of a light beam. The angular dispersion describes the dependence of the deflection angle θg(λ) on the wavelength for the grating. For the FPI spectrograph, the incidence angle dependence of the FPI transmission spectrum is used to separate the different spectral channels (see Sect. ). Therefore, we regard the incidence angle α as equivalent to the deflection angle θfp(λ) for the FPI.

For a blazed grating with a given ruling distance rg operated in the mth order and a Littrow-type spectrograph set-up (incidence angle and dispersion angle are as equal as possible), the relation of the wavelength and deflection angle θg (which equals the gratings blaze angle in this case) is given by see e.g. 7mλ=2rgsin⁡θg; consequently, 8dλdθg=2rgmcos⁡θg. A close to ideal choice of the ruling distance of the grating for a given wavelength is rg≈mλ. For a typical value of θg=30∘, a small wavelength shift by the width δλ of one ILF (or one spectral channel) changes θg by 9δθg≈0.58δλλ. For the FPI, the angle dependence (for a small incidence angles) is given by Eq. (): 10dλdθfp=-λθfp. The same small wavelength shift by one spectral channel δλ changes θfp by 11δθfp≈1θfpδλλ. This means that the angular change δθg for a single spectral channel of the GS is approximately given by its inverse resolving power, whereas for low FPI incidence angles, the angular change δθfp for a wavelength change of δλ can easily be 2 orders of magnitude larger than its inverse resolving power (e.g. factor of 100 for θfp≈0.6∘).

In either type of spectrograph, the angular deflection is translated to a spatial separation δx on a detector array via the imaging optics with focal length f (see Fig. b or f2 in Fig. a): 12δx≈fδθ. The desired spatial interval per spectral channel on the detector depends on the pixel size and the spectral sampling. Assuming that the ILF is sampled by five pixels of 10 µm pitch, this interval would be δx=50µm. Given that the size of a spectrograph is of the order of its focal length and its volume and mass scale with its third power see e.g., the above relations reveal the principal difference between the GS and FPI spectrograph in terms of size and resolving power (see Fig. a). For instance, one could argue that easily portable tools for humans have the size of a human hand (i.e. ca. 10 cm), which is about the size of a CubeSat miniature satellite see e.g.. The resolving power of the corresponding GS is about 1000 and, thus, quite close to that used by moderate-resolution DOAS measurements. The resolving power of the corresponding (f=10cm) FPI spectrograph is in the range of 105 and, therefore, capable of resolving individual rovibronic absorption lines of trace gases in the UV and visible spectral range.

These considerations point towards the advantages of FPIs for high-resolution spectroscopy, where they have been widely in use for more than a century see. Moreover, we have illustrated that the fundamental differences between grating and FPI result in different instrument sizes (or levels of mobility) for a given resolving power. However, these considerations do not yet include the spectrograph's light throughput and, hence, the maximum achievable SNR, which is also decisive for most atmospheric remote sensing applications.

In the following, we derive the general relationship between the sensitivity of a spectroscopic measurement and the light throughput of a spectroscopic instrument. The light throughput kH of a spectrograph defines the conversion of incoming spectral radiance I (in units of [photonss-1mm-2sr-1nm-1]) to a flux Jph,H of photons with energies (or wavelengths) from within a single spectral channel of the spectrograph (see Eq.  below).

The upper limit for the SNR of an atmospheric remote sensing measurement is often determined by photoelectron shot noise, i.e. by the number Nph=Jph,H⋅δt of photons detected within an exposure time period δt (defining the measurement interval). The noise of such a spectrum is given by Nph; thus, the photon SNR Θ of a spectrum can be approximated by 13Θ≈NphNph=IkH(δλ)δt. This can be translated to the corresponding limits ΔS for the detection of trace gas column densities using the effective differential absorption cross sections σ‾(δλ), which, in many cases, are a function of spectral resolution (compare Fig. ): 14ΔS≈1σ‾(δλ)Θ=1σ‾(δλ)IkH(δλ)δt. Here, the crucial role of the light throughput of the instrument becomes obvious, especially when the radiance of the light source (e.g. scattered sunlight) and the exposure time (e.g. time constant of the process to be studied) are fixed. Moreover, the choice of δλ is a compromise between optimal sensitivity (i.e. σ‾, typically decreasing with increasing δλ) and optimal light throughput (typically increasing with increasing δλ; see below). Particularly for trace gases with absorption cross sections consisting of discrete lines (e.g. OH, water vapour, or O2), the sensitivity increases almost linearly with the spectral resolution as long as it is much lower than the line width (see Appendix ).

When broadband light sources are used, a linear dependency of the light throughput on δλ is introduced. For line emitters where the spectral width of the emitted line is smaller than δλ (e.g. atomic emission lines), this is not the case compare e.g.. Here, we regard light sources that are broadband compared to δλ (scattered or direct sunlight or incoherent artificial light sources); therefore, we include the factor δλ in the light throughput quantification. Furthermore, the light throughput depends on the geometric beam acceptance of the optics (i.e. its étendue EH), which often introduces a further δλ dependency (see the following subsections). The spectrograph's étendue for a given spectral channel is approximated by the product of surface area AH and the solid angle ΩH of the corresponding light beam: 15EH≈AH⋅ΩH. Losses at the optical components are accounted for by a factor μ. From these effects, the light throughput can then be calculated as follows: 16kH=Jph,HI=μδλEH(δλ).

In the following, we compare the light throughput of FPI spectrographs with that of GSs for a given spectral resolution δλ. The losses at the optical components depend on their number, type, and quality. We assume that μ (accounting for these losses) is always optimised and that, apart from the OSM (introducing about a factor of 2 difference), there is no substantial difference in μ for the FPI spectrograph and GS. Thus, the light throughput is essentially determined by the étendue EH of an individual spectral channel.

We derive the étendue EH of GS and FPI spectrograph by approximating the surface area on the focal plane detector that is illuminated by light from a single spectral channel. The spectrograph's imaging optics determines the corresponding beam solid angle.

Imaging magnification does not affect the étendue (which is one of the reasons why the étendue is a universal measure of a spectrograph's quality), as it only converts a solid angle into surface area and vice versa. Therefore, for a light throughput comparison, we can ignore magnification and always assume ideal 1:1 imaging (i.e. collimating and focusing optics with the same focal length).

Investigating the light throughput per wavelength interval δλ allows the comparison of spectrograph set-ups with respect to their photon shot noise-limited SNR.

FPI spectrographs offer a way to reach large resolving powers with a largely reduced impact on the SNR (compared with GSs) while maintaining a mobile instrument set-up. This might allow substantially lower detection limits for trace gas measurements in the near-UV to NIR spectral range or may increase the measurements' spatial or temporal resolution.

When regarding noise-limited trace gas detection limits (as introduced in Eq. ), we find that the effective differential absorption cross section (and, thus, the sensitivity of the measurement) increases with spectral resolution for many gases in the near-UV to NIR regions. For absorbers with discrete lines (e.g. OH, water vapour, or O2), the sensitivity increase will be almost linear to the increase in spectral resolution (see Appendix , i.e. for our example a factor of ca. 250). For such gases, this effect outweighs the effect of reduced light throughput (0.01 compared with moderate-resolution GSs; Table ), and the corresponding noise-limited detection limits of the FPI spectrograph with interferometric OSM will be reduced by a factor of (250⋅0.01)-1=0.04 (0.4 for a grating OSM) compared with that of common, moderate-spectral-resolution DOAS measurements. By reducing the temporal resolution of FPI spectrograph measurements by a factor of 100 (i.e. increasing the exposure time, e.g. from 30 s to 50 min), the same photon SNR as that of moderate-resolution DOAS measurements (with 30 s exposure time) can be reached, reducing the detection limits by another order of magnitude.

In addition, the increase in sensitivity comes with a massive increase in selectivity for the following reason: on the one hand, the high spectral resolution allows one to use much more specific absorption structures for gas detection; on the other hand, the high spectral resolution reduces or removes the influence of undersampled Fraunhofer lines for sunlight measurements. Thus, detection limits can further be significantly lowered with respect to moderate-resolution measurements, which are, in many cases, also limited by cross interferences see e.g.. Consequently, line broadening effects could also add valuable information to retrievals of vertical atmospheric trace gas distributions, and the feasibility of distinguishing trace gas isotopologues is strongly improved. Besides improving water vapour isotopologue quantification see e.g., the separation of 34SO2 in volcanic emissions could also be possible using the differences in the absorption cross section, which are on a sub-nanometre scale e.g. and, thus, impossible to resolve with moderate spectral resolution.

Similar advantages are expected for the passive quantification of solar-induced fluorescence of chlorophyll by in-filling of narrow solar Fraunhofer lines with increased spectral resolution see e.g..

The following simple example outlines the impact that FPI spectrographs might have on atmospheric sciences. According to the above assessment, a high-resolution FPI spectrograph records a spectrum with a SNR Θ of 3333 with about a 1 h integration time. For scattered sunlight measurements in the UV, the tropospheric light path L can reach about 10 km. The absorption cross section of OH σ‾OH at around 308 nm reaches about 1.5×10-16 cm2 per molecule see. The detection limit of OH concentrations ΔcOH (see Eq. ) would then be 27ΔcOH≈ΔSOHL=1Θσ‾OHL=2×106molec.cm-3. This is already in the range of tropospheric background OH concentrations see e.g.. This detection limit can be lowered further by using active light sources like light-emitting diodes (LEDs) or Xe lamps instead of scattered sunlight or by using larger FPIs or arrays of parallel FPI spectrographs.

Furthermore, as assessed in Sect. , FPI spectrographs are expected to have similar advantages over FTS and GS measurements in the NIR. Thus, FPI spectrographs could also substantially improve remote sensing measurements of greenhouse gases (e.g. CO2 or CH4) or CO in Earth's atmosphere. Instead of the large spectral coverage with high resolution reached by FTS, several FPI spectrographs could record spectra in different spectral windows that are relevant for the trace gas retrieval e.g. an additional spectral window for O2 light path information; see e.g..

An important aspect with respect to the named and quantified benefits of FPI spectrographs is that the low level of complexity and the high mobility of presently used moderate-resolution GS measurements is maintained.

As a proof of concept, we built a prototype of an FPI spectrograph with a grating OSM at the Institute of Environmental Physics in Heidelberg (see Fig. a). It operates at around 308 nm. An FPI with high finesse (ca. 95) across a CA of 5 mm and a resolving power of ca. 148 000 (supplied by SLS Optics Ltd) was used with a compact OSGS. We recorded a spectrum of light from a UV LED that traversed a burner flame (see Fig. a) containing large amounts of OH typically several thousand parts per million; see e.g.. For a light path of about 1 cm, this leads to optical densities >1 for many OH lines see the OH absorption spectrum in Fig. , which is slightly altered due to the high temperature; see . Figure b shows the corresponding spectrum recorded by the FPI spectrograph prototype. The bright vertical stripes originate from a slight overlap of the individual FPI orders and, thus, also indicate their boundaries (compare Fig. b and c). The dark spots correspond to individual OH absorption lines. This is verified by calculating the intensity distribution using an instrument model and OH absorption data from . The orange box in Fig. b shows the region of the spectrum that is modelled in Fig. c. The locations of the individual OH absorption lines (dark spots) are clearly reproduced by the model, confirming the high resolving power.

Compared with the FPI spectrograph assumed in Sect. , the light throughput of this prototype instrument is reduced due to its smaller CA (i.e. by a factor of about 25; see Eq. ). The mobility advantages of FPI spectrographs as derived in Sect.  are already demonstrated by this still rudimentary prototype. Its volume is below 8 L, and it weighs less than 5 kg. The FPI can be replaced by an FPI with a larger CA without significantly impacting the instrument size.

A comprehensive description of this and further prototype instruments as well as the instrument models would go beyond the scope of this work and will be the topic of future publications.