Frequently the Czerny–Turner (Czerny and Turner, 1930) design is employed as sketched in Fig. 1a. However, other designs, e.g. imaging grating spectrographs, are also in use (see, for example, General et al., 2014, or Ferlemann et al., 2000; see Fig. 1b). The considerations presented in the following are largely independent of the particular spectrograph design. We also note that modern spectrographs using focal plane detector arrays, which simultaneously record the intensity of the entire spectrum of interest, enjoy the “multiplex advantage” over a scanning spectrograph. Therefore, using interferometers, which may (see Fellgett, 1949) or may not (see Barducci et al., 2011) feature a multiplex advantage, instead of grating spectrographs will probably not lead to better light throughput per se.

We assume a spectrograph entrance slit with width wS and height hS; thus an area AS=hS⋅wS (see Fig. 1); also we assume the aperture solid angle to be Ω. The étendue E of the instrument is thus given by 1E=Ω⋅AS=Ω⋅wS⋅hS. Let us consider a modern compact spectrograph (as, for example, described by General et al., 2014) with an entrance area (i.e. width×height of the spectrograph entrance slit) of AS≈0.6 mm2 at an F number of 4 (see definition of the F number in Eq. 6 below), equivalent to Ω≈0.05 sr. The total étendue (product of free entrance area and solid angle of acceptance of the entrance optics) E=AS⋅Ω of such an instrument would be about 0.03 mm2sr (or 3×10-8 m2sr; see also Sect. 2.3 below).

We further assume the spectrograph to be equipped with a linear detector array, with pixels of width wPix and a pixel “height” (pixel dimension perpendicular to the dispersion direction) sufficient to collect all light. The spectral interval δλ covered by a detector pixel will then be given by the spectrograph's linear dispersion dx/dλ and wPix as δλ=(dx/dλ)-1⋅wPix. Note that the spectral interval of a pixel is typically smaller than the spectral resolution of the instrument by a factor of 2 to 6.

Measurements of the clear-sky photon flux F at 320 nm indicate F≈20  mWm-2sr-1nm-1 (e.g. Blumenthaler et al., 1996) at a 30∘ observation elevation angle and at a solar zenith angle of 68∘.

The corresponding number of photons registered per pixel and second by such a spectrograph is given by (see, for example, Stutz and Platt, 2008, or Platt et al., 2015) 2ΔNPΔt=E⋅F⋅δλWPhot, where WPhot denotes the energy of a single photon (about 6.4×10-19 J for λ=320 nm). Assuming a typical δλ≈0.1 nm and the values for E and F from above results in about 108 photons per pixel and second.

In the following, we investigate measures to improve the spectrograph light throughput.

In principle improving the quality of the optics (reflectivity of the mirrors, grating efficiency, etc.) will increase the light throughput; however, typically the instruments are rather optimized in this respect, and the possible gain due to these measures is rather small (say of the order of 2). Moreover, improved optics can be combined with all measures to be described below. Therefore we restrict our discussion to other measures.

Overall, there are the following options (which to some extent can be combined):

Scale the size of the spectrograph, i.e. all three dimension lengths L1, height L2, and width L3, and thus the entrance slit area, while keeping the acceptance angle (i.e. the spectrograph F number) constant. We refer to this option as “spectrograph size scaling”.

If we wish to keep the light throughput constant when scaling (down) the size (given by L) of the instrument, we can just use a large number of (ideally) instruments with identical properties in parallel. The spectra of all instruments are then co-added so as to keep the light throughput constant.

Since E∝L2, we need to increase the number of individual spectrographs if L<L0. The number NSp of spectrographs required (which of course needs to be rounded to the nearest integer) will be 15NSp=L0L2. The total mass of an array of spectrographs scaled to L<L0 is then given by 16M⋅NSp=M0LL03⋅L0L2=M0LL0∝L. This means that the mass (and volume) shrinks with the scaling if, for example, a single spectrograph with characteristic dimension L0 is replaced by an array of N smaller spectrographs, each one scaled down in its linear dimensions to L0/N.

Thus, it appears that it would be of advantage to use a large number of very small spectrographs in order to reduce volume and weight of an instrument. However, there are limits to how far we can shrink a spectrograph (see also Sect. 3.3), at least as long as we consider conventional spectrograph design.

In the above section, we assumed that the spectrograph mass scales with the cube of the outer dimension, i.e. a characteristic dimension L. But how will the rigidness of such an instrument change if all dimensions are scaled by the same factor L/L0?

If, for simplicity, we assume the spectrograph to behave like a bar with length L, width w, and height h (see sketch in Fig. 5) on which an external force acts, then we can apply the famous case of bending a bar, which is described in most physics textbooks (see, for example, Meschede, 2015). When scaling the initial length L0 of the bar to some other length L by a factor L/L0 and likewise w0 to w=w0⋅L/L0 and h0 to h=h0⋅L/L0, we can calculate the scaling of Δh since 17Δh∝L3,Δh∝h-2,Δh∝w-1. Since w and h are scaled proportional to L, we have 18Δh∝L3L2⋅L=const. Thus the bar, in this case the spectrograph casing, will bend by the same absolute amount when subjected to a certain force. We can assume that the bending force actually scales with L as well because, for instance, thermal stress as well as external stress, for example, due to bending of the mounting base plate, is proportional to the dimension L. If we assume Hooke's law to hold, we can conclude that the deformation Δh of a spectrograph frame probably scales with 1/L. This, again, means that the performance of a scaled spectrograph will not change with scaling since the requirements for alignment of the optical elements also scale with L. For instance, in a scaled down spectrograph, the pixel size of the detector array will also shrink.

In conclusion, we can say that in first approximation, scaling of a spectrograph by changing all dimensions will not change its performance as far as it is determined by the geometry of the instrument, and therefore its mass will scale with L3 as assumed above.

Another point may be the thermal stability, which is a function of the thermal time constant of an object. The thermal time constant is given by the ratio of heat flow, being proportional to the surface of an object, and its heat capacity, which is (for a given material) proportional to the volume of the object. Thus the thermal time constant of a spectrometer should scale with 1/L.

Moreover, it can be assumed that a smaller instrument requires shorter and thus lighter wiring (although it amounts to only a small fraction of the total mass anyway).

Obviously, we cannot shrink spectrographs indefinitely since then they will not function any more. In addition to possible mechanical constraints, the following phenomena (see also, for example, Avrutsky et al., 2006) limit the shrinking of spectrographs:

Light diffraction at the shrinking entrance slit will degrade the spectral resolution.

Here we have a quick look at the effect of spectrograph-system optimization on the stray light level. Stray light can have negative effects on the precision of spectroscopic trace gas measurements, as, for example, pointed out by Platt and Stutz (2008). Note that stray light can be comparatively high in spectrographs, filtering a relatively broad wavelength interval from a continuous spectrum as in typical DOAS applications. As also pointed out by Platt and Stutz (2008), a typical stray light level of ISL/I≈10-5 as derived by illuminating the instrument with a monochromatic source (see, for example, Pierson and Goldstein, 1989) translates into stray light levels being closer to 10-2 than to 10-5.

Sources of stray light include light scattered by the optical elements (grating, mirrors, and the detector surface) of the instrument, reflection of unused diffraction orders off the spectrograph walls, reflection of unused portions of the spectrum from walls near the focal plane, and reflections from the detector surface (see, for example, Pierson and Goldstein, 1989). A further, potentially important source of stray light is due to incorrect illumination of the spectrograph: if the F number of the illumination exceeds that of the spectrograph, radiation will overfill the collimating mirror and hit interior walls of the instrument, from where it may be reflected to the detector.

In general, the amount of stray light is proportional to the ratio of the area of the scattering surfaces and the detector area; i.e. basically its amount scales with the inverse of the F number. Overall, it appears that the relative amount of stray light should not change when the spectrograph is scaled, such that its aspect ratio (and thus its F number) remains unchanged (i.e. according to case 1 in Table 1). Of course, running an array of spectrographs of identical design in parallel (case 4 in Table 1) should also not affect the relative amount of stray light.

It can be desirable to have a small or vanishing polarization sensitivity of spectrometers used for the analysis of sunlight reflected from Earth's surface or scattered in the atmosphere. In some satellite instruments, e.g. OMI and TROPOMI, “polarization scramblers” are used to reduce the polarization sensitivity of the spectrometer. There are many different designs of polarization scramblers (e.g. Lyot depolarizer or wedge depolarizer), which are based on plates consisting of birefringent material being placed in the optical path of the instrument; see, for example, Caron et al. (2012). These devices have in common that they are rather small plates consisting of two wedges (wedge angle around 1∘) made of birefringent material. These are placed in the optical path of the instrument, typically at a suitable position between the telescope entrance and the entrance slit. In the case of TROPOMI, two pairs of wedges (quartz and magnesium fluoride, respectively) are used, and their volume is around 1 cm3 (Babic et al., 2019).

Obviously, for very small spectrometers and telescopes, the depolarizer will also be very small (a fraction of 1 cm3), thus adding negligibly (<3 %) to the volume and weight of the instrument.

Another point which is particularly relevant for satellite applications is the use of direct solar reference spectra. The usual means of obtaining these spectra relies on directing sunlight via diffuser plates into the instrument. This approach, however, faces some technical difficulties due to the spectral structures introduced by the diffuser plates (see, for example, Richter and Wagner, 2001). Therefore, we recommend applying the “reference sector method” or related techniques, which are frequently used and do not rely on diffuser plates.

We also note that smaller instruments require shorter power (or data) connection wires; thermal ducts can also be shorter and thus lighter (see also Sect. 3.2). Similarly, radiation shielding (of e.g. detectors in satellite instruments from cosmic radiation) becomes much lighter for smaller instruments. Alternatively, one might want to keep the mass and thus the power lines and shielding constant and benefit from an enhanced light throughput. Moreover, using a large number of spectrometers in parallel has the potential advantage that the data from individual detectors, which were affected by cosmic radiation, can be sorted out and are not co-added during the evaluation procedure.

Finally, a spectrograph array design also reduces complexity since a much simpler optical design can be used for each pixel (or small number of pixels), which is just repeated many times (see, for example, Sect. 4.1). In fact, in most cases, a single (or a few) rather complex instrument(s) are replaced by a large number of comparatively simple instruments of identical design. There will likely be additional effort in cross-calibration, which, however may be offset by a simplified design.

In principle, this is a straightforward task: if all individual spectrographs of an array (i.e. set of spectrographs with identical spectral ranges and viewing directions) were truly identical in spectral resolution and spectral registration (wavelength calibration and dispersion), then the detector output signal of corresponding pixels only had to be individually digitized and co-added. How well this prerequisite for simple co-adding is actually met depends on the manufacturing process and its precision for the individual (miniature) spectrographs. If the deviations of the individual spectrographs only amount to a fraction of a pixel, one might choose to still simply co-add the spectra and accept a certain degradation in spectral resolution.

If it should be found that the individual spectrographs have considerable individual deviations in spectral registration, a correction by shifting and stretching/compressing the individual spectra prior to co-adding might be necessary. These tasks require some effort in post-processing; however with the rapid advancement of electronics and information technology in recent decades, this should not be a major problem. For instance, advanced bus systems could be used to interconnect the individual spectrographs.

For satellite instruments (see Sect. 4.1, below), the idea is to basically have one (or a small number of) spectrometer(s) per viewing direction. In the case of a single spectrometer (with a 1-D detector) per viewing direction, there would be no change in the amount of data generated compared to an approach using a single spectrograph with a 2-D detector (as, for example, in the OMI or TROPOMI instruments). In designs where several spectrometers observe the same ground pixel (see Sect. 4.1), data could be co-added on board. Thus, again there would be no increase in data rate compared to a single spectrograph with a 2-D detector approach.

Clearly, the widespread use of arrays of large numbers of (micro)spectrographs hinges on efficient manufacturing techniques for these instruments. Miniaturized spectrographs based on conventional spectrograph design are described by a number of authors, e.g. Avrutsky et al. (2006), Wilkes et al. (2017), and Danz et al. (2019). These authors also mention modern manufacturing techniques.

In particular, at present, technologies for mass production are available, like 3-D printing or automated machining of the frame. Also, the optical alignment of spectrographs can be automated; here replica optics could help. In addition, the required electronics and detectors have become very affordable during recent decades.

Furthermore, in the case of satellite instruments, space qualification and documentation of a large number of identical (and rather simple) spectrographs may mean less effort than that of a single or a few (relatively complicated) spectrographs.

A number of ideas for unconventional spectrograph designs were reported. For instance, how to enhance the light throughput of imaging spectrographs (for the visible and near-IR spectral range) is presented by Chrisp et al. (2020). Park and Choi (2013) suggested Fresnel optics to miniaturize spectrographs. Furthermore, a completely new principle for spectrograph design was proposed, for example, by Grundmann (2019a, b); it relies on using a special type of diode array as the only element of the instrument. The pixels of the diode array are manufactured in such a way that the bandgap of the semiconductor increases with the pixel number (this is achieved using a binary or ternary semiconductor with a composition varying with the pixel position). The light enters along the long axis of the detector array (which acts as a waveguide) at pixel 1, which has the smallest bandgap and therefore absorbs the longest wavelength radiation while transmitting radiation with shorter wavelength. Pixel 2 has a slightly wider bandgap absorbing radiation with slightly (by Δλ) shorter wavelength and so forth. The resolution of the device is approximately equivalent to Δλ. In a practical device, a spectral resolution of 0.01 eV at 3.5 eV (corresponding to about Δλ≈1 nm at ≈355 nm) was reached.

There are a number of satellite instruments in orbit (e.g. GOME, GOME-2A, B, C, SCIAMACHY, ODIN, OMI, TROPOMI, OMPS NM, GEMS (geostationary), EMI) which are based on small UV–visible–near-IR spectrographs (typical focal length around 200 mm) coupled to small telescopes (typical diameter 1 cm). The typical telescope field-of-view angle (for 1 ground pixel) is around 0.25 to 1∘.

Usually per wavelength range, one spectrograph is used. The total number of spectrographs is two (OMI; Levelt et al., 2006), four (TROPOMI; Veefkind et al., 2012; Dobber et al., 2006), four (GOME and GOME-2; Burrows, 1999), and eight (SCIAMACHY; Burrows and Chance, 1991; Goede et al., 1991; Burrows et al., 1995; Bovensmann et al., 1999).

The scanning (i.e. cross-track spatial resolution) is either achieved by a mechanical scanner or by imaging spectrographs (2-D spectrographs, where one dimension is devoted to wavelength, the second to space); see, for example, Levelt et al. (2006) or Veefkind et al. (2012). In particular, these imaging instruments are very sophisticated designs featuring extreme properties like very large cross-track fields of view combined with extremely small along-track aperture angles. These truly remarkable features come at a price: in some cases aspherical (or even free-form) optics have to be used, and only rather large F numbers are possible.

In order to reduce the weight and volume of instruments of this type, the single spectrograph (per wavelength range) could be replaced by an array of scaled down spectrographs, each observing 1 or a few ground pixels. Each spectrograph would have its own telescope; thus cross-track resolution could be achieved by aligning the field of view of the individual spectrographs accordingly, as sketched in Fig. 7a. Thereby the advantages of avoiding scanners by using the pushbroom principle are combined with a rather simple design (of the individual telescopes).

In fact, there could be one or several spectrographs per viewing direction and wavelength interval. This approach would have no more drawbacks, for instance, with respect to “destriping” measures, than existing whisk-broom designs (like OMI or TROPOMI). The cause(s) for the “striping phenomenon” (slight changes in the derived slant column density (SCD) values across the swath) are not fully understood, but it is probably due to somewhat different instrument functions for each viewing direction. On the other hand, such a spectrograph per viewing direction (SPVD) approach could have great advantages besides the obvious possibility of achieving better light throughput and thus SNR:

A much simpler spectrometer design. Here a conventional Czerny–Turner design or imaging grating design is assumed. Additional light throughput could be gained by the measures described in Sect. 2.3. Obviously the telescope has to be designed in such a way that the projection of the ground pixel matches the spectrograph entrance slit size (see Fig. 7b).

This latter effect (curvature of Earth) has the smallest influence on the ground pixel size at the edge of the swath. Thus a “flat Earth” approximation could be considered for the sake of simplicity.

In the following, we give two examples of possible satellite instruments based on arrays of microspectrographs. The hypothetical instrument designs are compared to the TROPOMI instrument, for simplicity, we only simulated the UV–Vis section of TROPOMI, but other wavelength ranges could be readily added. Relevant instrument parameters are summarized in Table 2:

One instrument (“Scaled 1”) has data similar to the TROPOMI UV–Vis section (Dobber et al., 2006; Veefkind et al., 2012; Babic et al., 2019), where the individual spectrographs are scaled down to approximately 1/10. For compensation, 100 spectrographs, each observing 6 ground pixels, would be run in parallel. The total étendue (≈0.065 mm2sr) of all spectrographs (variant A) would be somewhat smaller than the total étendue of the TROPOMI instrument (≈0.103 mm2sr). Therefore, we added another variant (B) of the instrument encompassing 200 spectrometer + telescope combinations (data given in square brackets in Table 2), arranged in two identical sets of 100 spectrometer + telescope combinations. In either case, each spectrograph would have its own (now very small; see Table 2) telescope. In the case of using 200 spectrographs (variant B), each set of 6 ground pixels would be observed by two spectrographs, thus doubling the étendue (to ≈0.13 mm2sr) and the signal, which would then exceed that of TROPOMI (alternatively, each spectrograph+telescope could observe only 3 ground pixels). Note that the total mass of the “scaled 1” instrument (not just one spectrometer) as given in the last line of Table 2 is about 1/100 of that of the TROPOMI instrument. This case also illustrates the design flexibility given by the spectrograph array approach.

The instrument (scaled 2) with 1 km by 1 km ground pixels throughout the swath with much (about 16.5) higher total étendue and comparable étendue per pixel (see Table 2) would provide a comparable signal-to-noise ratio as TROPOMI despite the 25 times smaller ground pixel area and could also feature constant ground pixel dimensions across the entire swath. For comparison, in order to achieve the same total étendue by just scaling up the instrument dimensions (e.g. from a TROPOMI-type instrument with M0≈100 kg) according to Eq. (5), a total instrument mass of ≈67⋅M0 or around 7 t would be required.

Obviously these are just examples to illustrate the potential of the new approach. Further spectrograph downscaling (for instance, to f=10 mm) would be possible. In addition, other combinations of spectrograph–pixel arrangements as well as the inclusion of further measures for improved light throughput (immersed gratings, imaging optics; see Sect. 2.3) are possible and must be explored.

MAX-DOAS spectrographs are typically equipped with miniature spectrographs (e.g. Ocean Optics, Avantes). Here, similar considerations apply as in the case of satellite instruments. For instance the typically used single spectrograph could be replaced by an array of scaled down spectrograph + telescope combinations as sketched in Fig. 6. In the simplest case, all spectrographs could point in the same direction, and the whole assembly would be tilted to measure at different elevation angles. Alternatively, the spectrograph + telescopes could point at different elevations, thus analysing the radiation at the chosen set of elevation angles simultaneously (see, for example, Leigh et al., 2007). While the latter approach would have the advantage that all elevations are observed truly simultaneously (as opposed to sequentially in the former approach), a problem could arise from slight differences in the instrument function of the individual spectrographs. Unlike the satellite case, there would be no natural way where all spectrographs see the same spectrum (e.g. by observing in the zenith direction).

In either case, one could argue that weight and volume of the spectrograph only constitute a small fraction of that of the entire MAX-DOAS instrument; however, the size of the instrument still scales with the spectrograph dimensions. Alternatively, the scaling could be used to enhance the étendue of the instrument and thus allow for proportionally faster measurements.

Another use of large arrays of spectrographs (+ telescopes) could be imaging applications where the usual need to make a compromise between spectral, spatial, and temporal resolution (see, for example, Platt et al., 2015) is removed or at least relaxed. For instance, an array of spectrographs (similar to the approach described by Danz et al., 2019) could be arranged with a spectrograph per image pixel in a compound eye (as found in insects) fashion.

Arrays of (miniature) spectrographs could also be applied in active long-path DOAS (LP-DOAS) instruments. In this case, a single, large telescope could be replaced by an array of small telescopes. As discussed above, the F number of these small telescopes would be about the same as in present instruments. If the total area covered by the telescope mirrors were the same, then there would be the same light throughput as in conventional active LP-DOAS designs. In this case, not only could the volume and weight of the spectrographs be reduced, but also the length of the telescope. This is because the F number of each small telescope remains unchanged (compared to traditional designs), while the diameter of the mirror (or lens) – and thus its focal length f – is scaled down.

Arrays of individual (largely identical) spectrographs could help to solve a number of design challenges for both satellite instruments and other applications.

Spectrograph arrays allow the Étendue and thus SNR to be improved independently from the spatial resolution.

We conclude that arrays of massively parallel spectrographs could solve the problem of achieving high light throughput with compact and lightweight instruments.

In particular, a reduction of the instrument volume and mass by 1 or 2 orders of magnitude at unchanged light throughput appears possible. This might be interesting for a number of particular design goals for satellite instruments:

Miniature satellites (e.g. CubeSats) could be equipped with spectrographs for Earth observation featuring sensitivity and spatial resolution comparable to present state-of-the-art instruments (like GOME-2, OMI, or even TROPOMI).

However, instrument properties, like radiometric accuracy and stability, initially and – in the case of 20 satellite instruments – over the entire mission, are rather questions of the particular instrument design. They have no direct connection to the question of how many individual spectrographs are used. Nevertheless, we are convinced that massively parallel miniature spectrographs are an attractive approach for future instruments.