Spaceborne remote sensing can be used to retrieve the atmospheric composition and complement the surface or airborne measurement networks. In recent years, a lot of attention has been placed on the monitoring of carbon dioxide for an estimate of surface fluxes from the observed spatial and temporal gradients of its concentration. Although other techniques may be used to estimate atmospheric CO2 concentration, the most promising for the near future is the absorption spectroscopy, focusing on the CO2 absorption lines at 1.6 and/or 2.0 microns. For this objective, the French space agency (CNES) has developed a new spectrometer concept that is sufficiently compact to be placed onboard a microsatellite platform. The principle is that of a Fourier Transform Spectrometer (FTS), although the classical moving mirror is replaced by two sets of mirrors organized in steps. The interferogram is then imaged on a CCD matrix. The concept allows a very high resolving power, although limited to narrow spectral bands, which is well suited for the observation of a few CO2 absorption lines. The laboratory model shows that a resolving power of about 65000 is achieved with a signal to noise on the spectra around 300. A modulating plate on the light path allows an easy change of the path difference. Although this component adds some complexity to the instrument, it greatly improves the information content of the measurements.
© 2010 Optical Society of America
There is an urgent need to understand the processes that control natural sources and sinks of carbon dioxide (CO2). Indeed, ocean and land masses currently absorb half of the carbon dioxide that is emitted through anthropogenic activities, slowing the greenhouse effect increase, but there are worries that this sink may slow down, or even reverse during the century. Our knowledge on current sources and sinks is hampered by the low density of observations of the atmospheric concentration. In this context, the scientific community has recognized the need for a spaceborne instrument that would monitor, at the global scale, the concentration of greenhouse gases. Although thermal infrared spectroscopy may provide information on the atmospheric CO2 concentration, its sensitivity is limited to the upper layers of the troposphere, which makes it ill-suited for the monitoring of carbon sources and sinks. Rather, the best suited techniques appears to be the differential absorption spectroscopy (DOAS) . This technique consists in quantifying the light absorption by carbon dioxide (CO2). For a given light path, the more the light is absorbed, the higher the CO2 concentration is. The light source is the Sun and its spectrum is measured after transmission through the atmosphere. The depth of the CO2 absorption lines in these spectra is representative of the CO2 quantity along the light path in the atmosphere.
Two satellite projects have been developed to launch by the US  and Japanese  space agencies. GOSAT was successfully launched in January 2009 and first products have been released although they do not show the necessary accuracy . The launch of OCO failed in February 2009 and the spacecraft was lost . The measurement technique consists in recording solar spectra in CO2 absorption bands in the short-wavelength infrared at 1.6 μm and at 2 μm. The accuracy threshold required for the measurement of the mean CO2 concentration is to retrieve the concentration at ± 1 ppm, corresponding to a relative accuracy of 0.3 % compared to the background concentration of 380 ppm.
The static Fourier transform spectrometer (FTS) developed by CNES  offers an alternative to perform the CO2 concentration measurement. The compactness of the instrument concept allows the monitoring of carbon dioxide with a micro-satellite rather than a mini-satellite. A micro-satellite leads to much reduced costs, and opens the potential for a constellation of such platforms for a dense coverage.
CNES has done a preliminary study to quantify the performances of the concept. An experimental breadboard was built for this objective. It is partially representative of what could be developed for a launch onboard a satellite and provides the information needed to evaluate the potential performances of a satellite-borne static FTS.
2. Instrumental concept of the static Fourier Transform Spectrometer
2.1. Brief introduction to Fourier transform spectroscopy
The Fourier transform spectroscopy provides an indirect measurement of the spectrum as the instrument records its interferogram. The interferogram is the interferometric intensity according to the path difference I(δ). Where δ is the path difference.
The spectrum is then retrieved by numerical processing of the interferogram through inverse Fourier transform.
A classical instrumental configuration for FTS uses a Michelson interferometer equipped with a moving mirror. The successive positions of the mirror sample the interferogram temporally. If the moving speed of the mirror is v, the path difference is δ = 2v·t and the interferogram is then a function of time (the interferometric signal recorded could be written I(t) as well as I(δ)). One can find-in depth descriptions of FTS in  or .
We now present the static configuration that was developed at CNES. It does not require the moving mirror of classical FTS, which reduces the overall size and mass of the instrument.
2.2. Instrumental concept
Figure 1 shows a diagram of the instrumental concept. One can recognize a Michelson interferometer with its classical components: a beam splitter generating two light beams, the two mirrors, a compensating plate to cancel out the chromatism of the beam splitter and a detector that measures the interferometric signal.
Note that other static FTS concepts have been studied for space applications. For example an imaging heterodyne spectrometer is considered for planetology studies by LESIA in Observatoire de Paris . Heterodyne spectrometers are similar to the spectrometer presented here in the sense that they focus on a narrow spectral band, which reduces the interferogram sampling constraints ,  and . Other examples of static FTS concept are described in , , , , ,  and .
2.2.1. Spatial sampling of the interferograms
Some key points make however this static FTS distinct from others. The main difference lies in the shape of the mirrors. The classical plane mirrors have been replaced by “stepped mirrors”. The two mirrors have a shape of stairs, one with tall steps, one with small steps. Figure 2 gives a picture of the mirrors used in the laboratory breadboard. The two stepped mirrors have complementary heights: the tall step height corresponds to the sum of the small steps. A spatial array of path differences is obtained by crossing the directions of the steps.
The stepped mirrors are built using a molecular bounding technique. A set of glass plates are assembled over one another with a small shift of typically 50 μm and 1100 μm (small and large steps respectively). The typical precision of bounding is 1 μm. There are around 22 plates for each mirror, which generates close to 500 samples of the path difference.
The stepped mirrors generate a spatial sampling of the interferogram. The interferometric signal depends on the position of the sample in the image of the stepped mirrors (it could be written I(x,y) as well as I(δ)). This instrumental configuration does not require a mechanism to record the interferogram as a function of the path difference, which is a significant advantage for a spaceborne instrument as a moving mechanism requires additional volume and mass. In addition, the measurements are highly sensitive to errors in the displacement of the moving mirror resulting from impure movement of the mechanism. Finally a motion mechanism induces a risk of breakdown particularly undesirable for spaceborne instruments .
The instrumental concept requires a detector that images the spatial variation of the interferometric signal so that a detector array substitutes for the classical monodetector. After the calibration, the image measured by the detector array provides the interferogram samples. The imaging system is shown as a single lens on the instrument sketch (Fig. 1) while it is actually more complex. The detector array contains many more times pixels than the number of interferogram samples. Each sample measured by typically 15×15 pixels from the detector array that we refer to as super-pixels. An image example recorded by the detector array is shown on Fig. 1. The colour of the different super-pixels stands for the interferometric intensity. This figure clearly shows the array of interferogram samples.
2.2.2. Narrow spectral band of measurement
A limitation of the measurement spectral bandwidth is necessary to respect the Nyquist-Shannon criterion for signal sampling. The criterion states that, for a spectral bandwidth ∆σ, the sampling frequency fs of the interferogram must be higher than fsmin = 2 · ∆σ  and . In the following, we use the sampling period written ∆sδ = 1/fs. The Nyquist-Shannon criterion for sampling is thus :
The sampling interval must be lower than a minimal interval ∆smaxδ = 1/(2∆σ).
Due to assembling constraints and limitations, the sampling interval cannot be lower than a typical value of 100 μm (corresponding to an elementary step height of 50 μm because of the light double path in the interferometer). Even if it were possible, a smaller sampling interval would suppose a reduction of the spectral resolution (by limiting the maximum path difference generated by the stepped mirrors). A sampling interval around 100 μm supposes a narrow spectral band of measurement of typically 50 cm-1.
The static FTS is consequently designed for spectral measurements in narrow spectral bands. Such spectral measurements can be well suited for the atmospheric sounding of one or two atmospheric species. In that case the spectrometer is strictly reduced to the need: the measurement spectral band is limited to the spectral region where the species are absorbing (the region where the sensitivity to the target species is located).
In the spectrometer, a narrow optical filter is placed in front of the detector array in order to limit the spectral band. It is represented on the diagram Fig. 1. The filter is built using the thin film technology widely used for narrow filters. Thanks to the spectral filtering, measurements can be done properly according to the Shannon-Nyquist sampling criterion as there is no undersampling.
The instrumental concept allows spectral measurements in a narrow spectral band. We now show that a high spectral resolution is possible thanks to the large maximal path difference.
2.2.3. High spectral resolution measurements
We use the full width at half maximum (FWHM) of the instrument line shape to quantify the spectral resolution. The spectral resolution of a FTS and its maximal path difference δmax are inversely proportional. The FWHM is defined by:
If we consider stepped mirrors generating 20 × 20 = 400 samples each 100 μm, the total amplitude of variation of the path difference would be 4 cm. By adjusting the number of samples, the height of the steps and the width of the measurement band, a maximal path difference around 10 cm can be considered. The static FTS can thus reach spectral resolution around FWHM ~ 0.06 cm-1. The maximal path difference of the experimental breadboard described latter is 6.3 cm (⇔ FWHM = 0.10 cm-1). Such a resolution is much higher than the resolution of the TANSO-FTS spectrometer onboard GOSAT-IBUKI (FWHM ~ 0.24 cm-1) or the resolution of the OCO spectrometer (~ 0.3 cm-1).
The spectral resolution is then better than that obtained with other static FTS using plane tilted mirror . Indeed, with a plane tilted mirror, a maximal path difference around 10 cm requires either very long mirrors (too long for a satellite borne instrument) or a tilt that leads to too diverging beams in the interferometer. The stepped mirrors concept allows parallel beams and a large path difference within a compact instrument.
3. Experimental model
To validate the concept, a laboratory model of the concept described above was built, assembled and tested at CNES, Toulouse. The full experiment design includes the spectrometer together with a calibration and test bench.
The spectrometer is composed of the “interferometric heart” and the imaging and detection system.
The experiment has been designed to perform several kinds of spectral measurements. Artificial light sources, a lamp and a tunable laser, can be observed in the laboratory. On the other hand, atmospheric spectra can be measured through the pointing to an external diffuser illuminated by the sun.
The wavelength range of the experimental spectrometer is centred around 1.6 μm (6360 cm-1), which corresponds to an absorption band of carbon dioxide. The bandpass filter of the spectrometer has been chosen to encompass around ten CO2 absorption lines. The transmission profile of the filter is presented later in Fig. 8. An atmospheric spectrum with the CO2 absorption lines visible is plotted in Fig. 12.
Table 1 sums up the main characteristics of the measurement spectral window of the spectrometer.
We now describe and discuss the various elements of the optical bench, with a focus on the interferometric heart, the imaging system and the light sources.
3.1. The interferometric heart
The interferometric heart has been assembled through molecular bounding for thermal stability reasons. It has been built by Winlight Optics that masters the molecular bounding technique. Figure 3 shows a sketch of the interferometric heart integrated in the experiment. Note that the beam splitter and the whole mirrors are also molecular bounded to a silica base.
The components are made of silica, a classical material for optical applications in the wavelength range around 1.6 μm.
The overall dimension are:
A specific component, hereafter referred to as the “modulating plate”, has been added to the instrumental concept previously introduced. Its is described in section 3.1.2.
3.1.1. The stepped mirrors
The tall step mirror is made of 19 steps. The small step mirror has 24 steps. The combination of the two mirrors generates the 19 × 24 = 456 samples of the interferogram. The height of the small steps is 80 μm which means a sampling interval ∆sδ = 160 μm. Each tall step is as high as the sum of the small steps. The height of each tall step is thus 24 × 80 = 1920 μm.
The path differences generated by the mirrors belong to the range [-6.3 cm;1 cm]. The interferogram has a bilateral symmetry between - 1 cm and 1 cm. The theoretical FWHM of the spectrometer is according to equation 2:
Two “calibration steps” are added to each mirror. These steps are just tilted in order to obtain a fine oversampling of the interferogram and thus to observe interfering fringes. This way, we expect a better monitoring of the interferometer drifts. The instrument may experience deformations due to thermal variations of its environment. The calibration steps are bounded at the edges of the mirror blocks. One located on the tall step mirror is visible on Fig. 3.
The mirrors define the pupil of the spectrometer. Each sample of the interferogram is obtained using the signal of only a fraction of the total surface. We define the sub-pupils or unit pupils included in the mirror surface. There are as many unit pupils as samples in the interferogram. The surface of the sub-pupils is 4×4 mm2. Each step is 4 mm wide. The overall dimensions of the pupil are given by the number of steps and their width:
- The transversal size is 20×4 = 80 mm (tall steps direction)
- The vertical size is 25×4 = 100 mm (small steps direction)
The total surface of the pupil of the spectrometer is thus 80×100 mm2. Table 2 lists the main features of the stepped mirrors.
Unfortunately, the molecular bounding technique does not allow a regular construction of the mirrors as the bounding accuracy is limited to a few microns. We measured a path difference error around 3 μm, which is very significant as the wavelength of measurement is around 1.6 μm. As a direct consequence, the stepped mirrors generate an irregular sampling of the interferograms. Yet, provided that the pathlengths are perfectly known, the spectra can be calculated although the processing requires an interpolation to obtain a regularly sampled interferogram.
3.1.2. The modulating plate
A specific component has been designed to ease the spectrometer calibration. The classical compensating plate is equipped with a mechanism that allows rotations around the vertical axis. By tilting the plate, the path difference can be slightly modified with a typical amplitude of one wavelength, which modulates the interferometric signal. The modulating plate also improves the measurements through the cancellation of the constant radiometric bias as is discussed in section 4.2 and in .
The design of the mechanism is rather simple: the compensating plate is mounted on a vertical rotation axis. Two piezoelectric actuators pushes on a lever arm linked to the rotation axis and set the plate in motion. They are located on the lower side of the interferometric heart base. More details about the component design and its benefits for spectra measurement are given in .
The main reason to chose the compensating plate, rather than a moving mirror, to modulate the signal is its tolerance to motion imperfections. Indeed, as it is used in transmission, the light direction is not affected by rotation error. On the other hand, a moving mirror would deflect the light beam were the translation not pure. Moreover, the rotation angle required for a path difference variation around the wavelength is quite small. This angle depends on the width of the plate and its optical index. In our case the width of the plate is 20 mm and it is made in silica (nSiO2 = 1 .44). The specification of the mechanism was to induce a maximal path difference variation equal to two times the working wavelength (2×1.6 = 3.2 μm). This can be achieved thanks to a maximal rotation of the modulating plate equal to 3 mrad. Finally owing to the narrow spectral window of measurement, a slight rotation of the compensating plate does not generate any significant chromatic variation between the two arms of the interferometer. The features of the modulating plate are given in Table 3.
The main application of the modulating plate is to calibrate the spatial gain of the detector array. This calibration is described in section 4.1.2. Besides, the mechanism offers the opportunity to increase the number of samples in the interferograms. Indeed, several static interferograms can be acquired for several positions of the modulating plate. These measurements increase the interferogram sampling and therefore the quality of the measurements. An optimal sampling of the interferograms is obtained with the acquisition of two samples spaced by λ0/4 for each path difference generated by the stepped mirrors. This supposes at least two interferogram measurements for two positions of the modulating plate. Moreover, the increase of the number of samples allows the increase, by the same ratio, of the spectrometer spectral window width. On the other hand if the spectral window is wide enough for the application, the use of the modulating plate allows a reduction of the number of samples that are generated by the stepped mirrors.
The complexification of the instrumental concept due to the implementation of the compensating plate is overcome by the benefits it brings:
- It increases the quality of the measurements. An optimal sampling helps particularly to reduce the errors induced by the irregular sampling of the stepped mirror.
- It reduces the constraints on the spectral window. The mirror configuration can be consequently simplified: fewer steps are required for a given spectral window.
3.1.3. Beam splitter coating
The design of the coating on the beam splitter posed specific challenges because of the demanding specifications:
- The coating has to divide equally the light on the whole spectral window which is typically about 6 nm wide at 1.6 μm central wavelength
- Its properties must keep uniform on the whole beam surface: ~ 80×100 mm2
- A uniform polarisation response was required
To ease the realisation, it was decided to set the light incidence on the beam splitter to 30º. The CILAS company defined the appropriate coating and its deposition technique. CILAS finally made the splitting coating according to the conclusions of the study. The deposition technique is ion beam assisted deposition. It enables to reach the harsh specifications. The resulting coating induces mechanical constraints onto the plate and makes it bend. The curvature was measured on a test sample. with a curvature that led to a flat surface with the coating deposit. The solution was to apply the coating to a silica plate that was specifically designed with a curvature that led to a flat surface with the coating deposit.
3.2. The imaging system
The imaging system is located at the output of the interferometric heart. It is made of an objective lens which projects the image of the stepped mirrors on a InGaAs detector array. The image of all the steps must be sharp on the detector. The depth of field of the imaging system must be equal in our case to ~ 36 mm.
The imaging system is made with two doublets. A diaphragm set at the focal point of the first doublet selects the field of view of the spectrometer, which is 7.6 mrad. For a satellite in orbit at 700 km this field of view corresponds to a spot on the Earth with a diameter equal to 5 km when viewing nadir. The imaging system is telecentric, which helps to have a correct image of the mirrors according to the depth of field (the magnification is kept constant).
The magnification of the imaging system is such that the surface of a spatial sample on the stepped mirrors corresponds to the surface of a “super-pixel” on the detector array. In our case the magnification is 1/10 in order to project the samples with a surface of 4×4 mm2 on super-pixels made of 13×13 pixels (the surface of a single pixel is 30×30 μm2).
To obtain the interferogram value for each step, the mean signal of each super-pixel is calculated. The pixels at the periphery of the super-pixels are not taken into account in order to avoid side effects (see the detector array image in Fig. 10). The image processing also applies the calibration corrections that are needed as is described in section 4.1.
Figures 4 to 9 show the different stages of a spectrum measurement for two light sources: a lamp and a laser. Figure 4 and Fig. 5 are the images recorded by the detector array using respectively a lamp and a laser as light source. Each square corresponds to a sample of the interferogram. This illustrates the spatial sampling made by the stepped mirrors. The colour of the squares stands for the interferometric intensity of each sample expressed in low significant bits (LSB). One can also distinguish the calibrating steps on the top and right sides of the mirrors. These steps show fringes representative of their tilts on the laser image (Fig. 5). The corresponding interferograms plotted according the path difference are given on Fig. 6 and 7 for respectively the lamp and the laser. Due to the spectral width of the lamp and the narrowness of the laser, the lamp interferogram (Fig. 6) is less contrasted than the laser interferogram (Fig. 7). This difference can also be seen directly by comparing the images (Fig. 4 and 5). The reduced fringe visibility seen on the laser interferogram at large path difference is a consequence of the finite range of angles in the interferometer field of view (self-apodisation). The various image processing steps (image averaging, corrections for calibration and inverse Fourier transform) lead to the incoming spectra such as those shown on Fig. 8 and 9 for respectively a lamp and a laser. These two plots give examples of measurements from the static FTS. The lamp spectrum (Fig. 8) is in fact the spectral profile of the bandpass filter while the laser spectrum (Fig. 9) represents the instrument function.
3.3. The light sources
Three different light sources may be used as input of the instrument: two artificial sources and a natural one. The artificial sources are a lamp and a tunable laser. They are both injected into an integrating sphere. The light from the sphere is collimated to simulate Earth observation conditions. The natural source is the Sun through the atmosphere. The sunlight is reflected into the laboratory and the spectrometer by a scattering reflector put in CNES garden, in front of the window of the laboratory.
The artificial sources are used for calibrations. The incandescent lamp is a spectrally wide and flat source. It is used to measure the transmission of the bandpass filter of the spectrometer. On the other hand the laser is a spectrally narrow light source. It enables to measure the instrument function such as the one shown in Fig. 11 in section 5.1. Laser measurements are also used to calibrate the path differences (see section 4.1.4).
4. Exploitation method
We now describe the data acquisition and processing steps designed for the static FTS. The calibration of the spectrometer must take into account the specificities of the instrument, and in particular the spatial sampling.
4.1. Calibration of the spectrometer
The different steps of the spectrometer calibration are similar to what is done usually for an optical instrument: the dark signal is subtracted, then the pixel gains are measured and finally the non-linearity of the detector is evaluated. One specificity of the concept is the use of a detector array, which requires to calibrate all detector pixels, and to evaluate the inter-pixel calibration. In addition, the detectors measure interferometric signal and the calibration method must be designed accordingly. Finally, the measuring concept requires a very precise knowledge of the path differences that must be therefore calibrated accurately.
4.1.1. Dark signal
The dark signal is measured by blocking the incoming light to the detector with a remote controlled shutter. A series of dark images are recorded and the mean image is derived. During this step, we also identify anomalous pixels that show an unstable or a particularly high or low dark current. These pixels are not used for further processing, and the dark current images are subtracted from the measurement images.
4.1.2. Spatial gain
The spatial gain calibration provides a map of the interpixel sensitivity. It is also called flat-field correction. Such calibration is widely done for imaging systems using a detector array. The classical method of calibration consists in projecting a uniform image on the detector. The image recorded by the detector (corrected from the dark signal) gives the sensitivities of the pixels. Once the calibration is done, the correction of the interpixel sensitivity is made by dividing the images from the detector by the flat-field image. However, this simple method cannot be applied here because the interfering beams generate a non-uniform signal on the matrix as shown in Fig. 4 above. As an alternative, we use several positions of the modulating plate as is now described.
Using the Wiener and Khintchine theorem in the case of small path difference variations around a given path difference δ, the signal at the output of the interferometer can be written:
The first term represents the constant part of the interferogram and the second stands for the useful interferometric signal. We introduced the maximum of the signal M, its minimum m and the mean wavenumber σ0. This representation of the interferometric signal for small path difference variations is all the more valid since the spectral interval remains narrow, which is the case for our spectrometer including a narrow passband filter. If the object is located at an infinite distance, irradiance on the mirrors is uniform. The constant part of the interferogram must be then the same for each sample. In that case, the differences in the constant terms seen by the detector are representative of the interpixel gains. To make the flat-field calibration, the interferences must be blurred in order to only keep the constant term of the interferogram. This is done using the modulating plate. The flat field correction can also be assessed by measuring the contrast of the interferometric signal for each pixel .
According to equation 4, the interferometric part of an interferogram is cancelled if the path difference varies by multiple of the mean wavelength λ0 = 1/σ0 during the signal recording. The mean of the cosine is null. For the static spectrometer calibration we calculate the mean image of 40 interferogram images obtained for 40 positions of the modulating plate. The positions of the modulating plate are such as to generate 40 path difference variations within an interval of which the size is equal to the central wavelength.
For the correction of the spatial sensitivity, the inverse and normalized image is used. Such an image is shown on Fig. 10. Note the global look of the image in which one can distinguish the “radiometric dome” (larger correction factors at the periphery and lower ones at the centre) typical for such image. This radiometric dome is due to the variation of the incidence of the light according to the position in the image. One can also notice the significant side effects at the interface between two steps produced by the sharp shape of the edges of the stepped mirrors. The pixels affected by these effects are not used for spectral measurements.
Preliminary measurements indicated that the InGaAs array used in our optical design has a highly non-linear response. It was therefore necessary to fully calibrate its response. The method consists in measuring a series of 200 images for a series of 200 incoming fluxes. The light intensity is set by the driven diaphragm of the lamp illuminating the integrating sphere. The corresponding flux is measured by a calibrated photodiode used as a reference.
The non-linearity calibration consists in measuring the instrument response according to the incoming flux. Then the response is fitted by a degree 7 polynomial. This polynomial is finally used to correct the measurements from the effects of non-linearity. The level of the polynomial was chosen empirically to get the appropriate precision of correction.
Note that the interferences may lead to rather low signal for some of the super-pixels, which is unfavourable for the non-linearity calibration. Indeed, the range of calibrated detector levels may be too small with respect to what is actually seen during the spectral measurements. A significant part of the detector response could therefore be uncalibrated. On the other hand, if the signal is minimal for a given position of the modulating plate, it is then maximum when the plate is rotated for a path difference variation of λ0/2. Two series of measurements for the two positions of the plate are therefore recorded. For each super-pixel, the series with the highest signal is used for the calibration.
4.1.4. Path difference measurement
The Fourier transform spectroscopy requires a highly precise knowledge of the path difference which must consequently be calibrated.
The measurement is based on a chromatic method. Interferograms are recorded for several emission wavenumbers σ of the tunable laser. We make 1600 interferogram measurements for 1600 different wavenumbers. The interferogram equation for a monochromatic source is:
The power of the laser B(σ) is monitored for each wavenumber and data are corrected from its variations. The interferogram can be written as a function of the wavenumber for a given path difference (a given super-pixel) :
For each super-pixel, the signal according to the wavenumber is thus a cosine with a frequency proportional to the path difference. The maximum M and the minimum m of the cosine are representative of the modulation of the signal at the considered path difference. A contrast is defined as a function of the path difference :
It is used during the data processing to take into account the auto-apodisation of the interferogram.
The 3 parameters (δ, m and M) are retrieved for each super-pixel applying a fit to the 1600 points corresponding to the 1600 laser wavenumbers. The precision of measurement of the path difference is estimated around 1 nm. Knowing that centimetres are measured, the relative precision is around 107.
Note that the data used for this calibration are first corrected from dark signal, spatial gains and non-linearity.
4.2. Interferogram processing
- The steps of the the interferometric image processing are:
- Subtracting the dark signal
- Correcting the spatial gains
- Calculating the mean signal on each super-pixel
- Correcting the non-linearity
- Calculating the spectra knowing the interferogram according to the path difference
The interferograms are actually obtained following a special measurement procedure: “4 phase measurement” involving the modulating plate. This supposes the recording of 4 interferograms (I1, I2, I3 and I4) with a phase-shift between 2 consecutive interferograms equal to π/2. A π/2 phase-shift is introduced by a rotation of the modulating plate corresponding to a path difference variation of λ0/4. The π phase-shifted interferograms are subtracted:
This way, the constant part of the interferograms is nulled, including the constant errors. Then the interferogram is obtained by the union of the two remaining interferograms:
This special measurement pattern improves significantly the quality of the spectra.
The explanation for the improvements are:
- The modulation cancels the constant errors
- The sampling resulting from the union of the two modulated interferograms is optimal.
The spectrum calculation is then easier. The interpolation of the interferogram on a periodical sampling pattern is indeed helped by the couples of λ/4 spaced samples.
The final interferogram has twice more samples than a single interferogram at the output of the interferometric heart. Its sampling is particular: it combines the step height (~ 160 μm) and the path difference variation induced by the modulating plate (λ0/4 ~ 0.4 μm). Such sampling allows to double the spectral window of the spectrometer compared to the case without modulating plate. The fact that the spectral window is doubled can be seen on the spectra such as in Fig. 8 and 12: the spectrum calculation interval is far larger than the filter profile which was originally defined for an interferometric heart without a modulating plate.
Here can be discussed the advantages and the drawbacks of the modulating plate in the interferometric heart. Apparently it complicates the instrument design and operation: it is quite risky to put a mechanism in a satellite borne instrument. This supposes a significant additional mass. On the other hand, beside the radiometric gain due to modulation, the modulating plate lets consider a simplified interferometric heart with fewer steps (for a given spectral window). Moreover, the mechanism can be much simpler than that of the mirror displacement used in classical FTS. Here, the component moved by the mechanism is quite tolerant to impure movements unlike mirrors. The on ground static FTS helped to prove the benefits of the modulating plate. The challenge would not be the same for a flight model but the pro and the con have to be studied to decide whether the modulating plate brings more advantages than disadvantages.
We present in the next section the results of the characterisation of the data provided by the experimental spectrometer.
5. Instrumental characterisation
The quality of the measurements of the static FTS is assessed according different criteria concerning different parameters. The spectral quality is studied through the resolution of the spectrometer. The radiometric quality is given using signal-to-noise ratios (SNR). SNR for the interferograms and the spectra are measured.
5.1. Spectral resolution
The spectral resolution is controlled by the maximal path difference δmax which determines the width of the instrument line shape (ILS). For an ideal FTS the full width at half maximum of the ILS is defined by equation 2. It corresponds to the FWHM of the sinc ILS function for an ideal spectrometer when the interferogram is not apodised.
To determine the spectral resolution of the spectrometer, we retrieve the parameters of the sinc ILS function when a monochromatic source (the laser) is observed. Using a laser spectrum, the measured points are fitted with the theoretical function:
with K a coefficient to adjust the level of the function, σ0 the wavenumber of control of the tunable laser and ε(σ0) the tuning error of the laser (the observed spectrum could be not strictly centred on σ0). The parameters retrieved by the fit are K, δmax and ε(σ0). The spectral resolution is deduced from δmax.
Figure 11 shows the measured points (red points) and the best theoretical sinc function (in blue). The retrieved maximal path difference is δmax = 6.23 cm whereas the maximal path difference measured thanks to the chromatic method presented in section 4.1.4 is equal to 6.27 cm. This difference implies a widening of the ILF of 1 · 10-3 cm-1, which is smaller than the widening due to the field of view (around 0.05 cm-1). The resolution degradation due to a non ideal instrument are not measurable with the given radiometric performances of the breadboard. For information the tuning error of the laser is ε(σ0) = -1.2 cm-1 for a control wavenumber σ0 = 6357.3 cm-1, which corresponds to an error ε(λ0) = 7 pm for a control wavelength λ0 = 1573 nm.
We use equation 2 to determine the spectral resolution of the spectrometer and we find FWHM ~ 0.098 cm-1, which results in a resolving power of ~ 65000. Such resolution and resolving power are only achieved in space by instruments such as TES , MIPAS  or SPIRE .
We gather several parameters describing the spectral resolution of the static FTS in Table 4.
5.2. Signal-to-noise ratio
This section is dedicated to the SNR performances of the static FTS. They are assessed for both the interferograms and the spectra. We present the conventions for calculation of the SNR and the main results.
5.2.1. SNR for the interferograms
We consider two kinds of SNR for the interferograms:
- The “spatial SNR” which quantifies the noise within the interferogram from a sample to another. The term spatial is used because the different samples are spatially spread on the stepped mirrors.
- The “temporal SNR” which is determined comparing two interferograms recorded for two different moments.
The spatial SNR is obtained by processing the standard deviation of a lamp interferogram for the high path difference samples. Such an interferogram can be seen on Fig. 6. For high path differences the interferogram remains constant because the incoming spectrum has not high-frequency variations. The fluctuations of the interferogram are therefore interpreted as a noise, which is quantified by the standard deviation of the samples at large path differences. This standard deviation (written Nspatial(Ihigh δ)) is compared to the constant part of the interferogram before modulation (written S). The spatial SNR is given by the equation:
The factor 2 is introduced because the amplitude of the interferogram is doubled by the phase modulation. Spatial SNR around 4100 is observed for the interferograms.
The temporal SNR is obtained by comparing the standard deviation of the difference of two consecutive interferograms (I1 and I2) and the mean signal S. It is defined by the equation:
Where Ntemporal(I2 - I1) is standard deviation of the difference of two modulated interferograms. As previously, the factor 2 is used to take into account the effect of phase modulation on the interferogram amplitude. The factor √2 is introduced because the variance of a difference is the quadratic sum of the variances (when considering independent noises). Temporal SNR around 6100 is measured for the interferograms.
The objective of the on ground breadboard was to reach interferogram SNR higher than 3000. This value was estimated when considering the precision goals for atmospheric carbon dioxide sounding. SNR better than 3000 would allow to measure carbon dioxide concentration with a precision around 1 ppm. The interferogram SNR objective is reached.
5.2.2. SNR for the spectra
The SNR objective for spectra is 300. As for interferograms two kinds of SNR for spectra are distinguished:
- The “spectral SNR” which quantifies the spectral noise into the spectra. The term spectral is used because we look for a perturbative signal affecting the different spectral samples.
- The “temporal SNR” which determines the level of random signal between two consecutive measurements (as for the interferograms).
The spectral SNR is obtained looking at a lamp spectrum such as the one shown in Fig. 8. The noise level is given by the standard deviation of the spectrum in the feet of the bandpass filter where signal is expected to remain null. This standard deviation is written Nspectral(Bfeet). The signal level (still written S) is taken as the maximum of the spectrum. The spectral SNR expression is:
Spectral SNR around 345 is observed for apodised lamp spectra. Note that if the instrument was photon noise limited this SNR would somehow overestimate the actual spectral SNR (the noise is assessed for a signal lower than the signal used for the SNR calculation). Yet, the breadboard is not photon limited and the method of calculation has been considered as giving an estimation of spectral SNR representative of the performances of the FTS.
The temporal SNR for spectra is calculated by carrying the difference between two consecutively recorded spectra (B1 and B2). The standard deviation of the difference is considered as the temporal noise. It is written Ntemporal(B2 - B1). The signal level S is the maximum of the spectra. The temporal SNR is:
As for equation 13 the factor √2 reflects the fact that the standard deviation of a difference is used. Temporal SNR around 1200 are obtained for apodised spectra. It is much higher than the spectral SNR, which lets consider that a part of the spectral noise is systematic. It could be due to a lack of precision on the path difference vector used for the spectra calculation.
These results indicate that the objective of spectrum measurements with SNR higher than 300 is reached with the present instrument configuration. Table 5 gathers the SNR results for the spectrometer.
6. Atmospheric measurement
The objective for a potential spaceborne spectrometer based on the static FTS concept would be to provide spectra for atmospheric sounding applications, in our case the sounding of atmospheric carbon dioxide. A simple way to explain the measurement of the carbon dioxide concentration is to consider that the concentration is deduced from the depth of the CO2 absorption lines in the atmospheric spectra. For a given observation geometry, the deeper the absorption lines of the spectrum the higher the concentration is.
Figure 12 shows an atmospheric spectrum (in solid red) measured on ground looking at the sun through the atmosphere. The shape of the spectrum is due to the bandpass filter (plotted on Fig. 8) and to the CO2 absorption lines. Solar Fraunhofer lines can also be observed. A simulated atmospheric spectrum is also plotted (dashed blue) as well as the residuals between the measurement and the simulation. The simulated spectrum is provided by a radiative transfer code developed for studying CO2 retrieval in the short-wavelength infrared . It can use the HITRAN  as well as the GEISA  spectral databases.
Atmospheric CO2 concentration measurements actually require a retrieval procedure based on inverse problem theory . The on ground breadboard can be used to attempt such measurements. Spectra such as the one presented in Fig. 12 could be recorded and a retrieval procedure could be applied to obtain the corresponding atmospheric CO2 concentration.
This paper presents a new instrumental concept for a Fourier Transform Spectrometer that is well suited for spaceborne atmospheric sounding. The interferogram sampling is achieved through two sets of stepped mirrors, instead of a moving mirrors and the image of the interferogram samples is acquired by a detector array. The instrumental concept allows high spectral resolution measurements over a narrow spectral window selected by a bandpass optical filter.
Based on the theoretical concept, a laboratory model was built in CNES Toulouse. The spectrometer measures spectra around a set of CO2 absorption lines close to 1.6 μm. A modulating plate that can rotate by a small angle to generate small variations of the optical path was added to the initial concept. A combination of measurements acquired with different orientations of the plate allows a calibration of the measurements. A well-chosen combination of measurements can be used to reduce some error sources through the cancellation of constant bias and the optimisation of the interferogram sampling.
The experimental spectrometer has been characterised. The spectrum SNR is better than 300. The resolving power is around 65000, which is comparable to that of currently flying or planned spaceborne high-resolution spectrometers. Yet, the mass and dimensions of the new concept could be compatible with a micro-satellite platform, which has a favourable impact on the mission cost.
The authors are very grateful to the three anonymous reviewers who provided helpful comments that led to a significant improvement to the paper.
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