Spectral performance analysis of the Aeolus Fabry-Pérot and Fizeau interferometers during the first years of operation

In August 2018, the European Space Agency (ESA) launched the first Doppler wind lidar into space which has since then been providing continuous profiles of the horizontal line-of-sight wind component at a global scale. Aeolus data has been successfully assimilated into several NWP :::::::: numerical ::::::: weather ::::::::: prediction :::::: (NWP) : models and demonstrated a positive impact on the quality of the weather forecasts. In order to :: To : provide valuable input data for NWP models, a detailed characterization of the Aeolus instrumental performance as well as the realization and minimization of systematic error sources is crucial. 5 In this paper, Aeolus interferometer spectral drifts and their potential as systematic error sources for the aerosol and wind product ::::::: products are investigated by means of instrument spectral registration (ISR) measurements that are performed on a weekly basis. During these measurements, the laser frequency is scanned over a range of 11 GHz in steps of 25 MHz and thus spectrally resolves the transmission curves of the Fizeau interferometer and the Fabry-Pérot interferometers (FPIs) used in Aeolus. Mathematical model functions are derived in order to analyze the measured transmission curves by means of non10 linear fit procedures. The obtained fit parameters are used to draw conclusions about the Aeolus instrumental alignment and potentially ongoing drifts. The introduced instrumental functions and analysis tools may also be applied for the upcoming missions using similar spectrometers as for instance EarthCARE (ESA) which is based on the Aeolus FPI design.

means of a beam expander and with that reducing its divergence to 555 µrad, the light is directed to the Fizeau interferometer which acts as a narrowband filter with a full width at half maximum (FWHM) of 58 fm (135 MHz) to analyze the frequency 90 shift of the narrowband Mie backscatter from aerosol and cloud particles. The Fizeau interferometer spacer is made of Zerodur to benefit from its low thermal expansion coefficient. It is composed of two reflecting plates separated by 68.5 mm leading to an FSR of 0.92 fm (2190 MHz), which is chosen to be 1/5 th of the FPI FSR. The plates are tilted by 4.77 µrad with respect to each other and the space in-between is evacuated. The produced interference patterns (fringes) are imaged onto the accumulation charge coupled device (ACCD) on different pixel columns, whereas different laser frequencies interfere on 95 different lateral positions along the tilted plates. The ACCD does not image the entire spectral range covered by the aperture but only a part of 0.69 fm (1577 MHz) which is called useful spectral range (USR). This so-called fringe imaging technique using a Fizeau interferometer (McKay, 2002) was especially developed for ALADIN (ESA, 1999).
The light reflected from the Fizeau interferometer is directed towards the so-called Rayleigh channel on the same beam path and linearly polarized in such a direction that the beam is now transmitted through the PBSB. The Rayleigh channel is based on 100 the double-edge technique (Chanin et al., 1989;Flesia and Korb, 1999;Gentry et al., 2000), where the transmission functions of two FPIs are spectrally placed at the points of the steepest slope on either side of the broadband Rayleigh-Brillouin spectrum originating from molecular backscattered light. For ALADIN, the two FPIs are illuminated sequentially by using the reflection of the first FPI (called direct channel or channel A) to illuminate the second FPI (called reflected channel or channel B). A conceptually similar approach was introduced by Irgang et al. (2002) and was adapted to the double-edge configuration for 105 ALADIN to gain higher radiometric efficiency for the Rayleigh channel. This arrangement also results in different maximum intensity transmissions for both FPIs, compared to a parallel implementation of the double-edge technique with equal filter transmissions. The two FPIs are manufactured by optically contacting the plates to a fused silica spacer with a plate separation of 13.68 mm leading to an FSR of 4.6 pm (10.95 GHz), whereas the spacing of the direct channel FPI is further reduced by a deposited step of 88.7 nm (one quarter of the laser wavelength) to shift its center frequency with respect to the reflected channel 110 by 2.3 pm (5.5 GHz). The space between the plates is evacuated. The plate reflectivity is measured to be 0.65 resulting in an effective FWHM of the transmission curves of 0.70 pm (1.67 GHz), whereas this value does consider defects on the plates as for instance their roughness, bowing and lack of parallelism. It does not consider any further modifications of the FWHM caused by the spectral characteristics of the light reflected from the Fizeau interferometer. This issue, and in general the shape of the interferometer transmission curves are discussed in more detail in Sect. 4.3. The light transmitted through the direct 115 channel and the reflected channel FPIs is imaged onto the same ACCD by a single lens after it was combined by an PBC with a small offset angle to 45 • , resulting in two horizontally separated circular spots. As the FPIs are illuminated with a nearly collimated beam of 1 mrad full angle divergence, only the central 0 th -order of the inference pattern is imaged onto the ACCD detector. As the FPIs are rather temperature sensitive (≈ 455 MHz/K which corresponds to ≈ 81 m/s/K), they are enclosed in a thermal hood to reach a long-term temperature stability of about ±10 mK. On the short time scale of a wind 120 observation (12 s), the temperature stability is even better than 3 mK, which translates to wind speed variations of less than 0.2 m/s. For the sake of completeness, the main specifications of the Fizeau interferometer and the FPIs are listed in table 1.
The values given above and as listed in table 1 are the essential design parameters and specifications. The actual spectroscopic performance and resultant operational parameters, as for instance the fringe width and shift, line profiles and measurement accuracies, are profoundly influenced by a multitude of optical and technical considerations. These include alignment accuracy 125 and stability, uniformity of plate illumination, spurious and parasitic reflections, detector non-linearities, and of course any changes or fluctuations therein on short and long time scales. The subject of this paper is thus the evaluation of the impact of these factors, based on detailed analyses of nearly three years of spaceborne data. In particular, data from ISR measurements that are performed on a weekly basis are used.
3 Instrument Spectral Registration (ISR) 130 The ALADIN instrument is able to perform special instrument modes that are used for instrument performance monitoring and calibration purposes. One of these modes is the so-called instrument spectral registration (ISR), which is used to characterize the Table 1. Specifications of the Mie spectrometer and the Rayleigh spectrometer of the ALADIN instrument (Reitebuch et al., 2009 Fizeau interferometer and the FPIs transmission curves and with that, to monitor the overall ALADIN instrumental alignment.
In the following, the ISR measurement procedure as well as the corresponding data processing steps are shortly elaborated.

Measurement procedure 135
During an ISR measurement, the laser frequency is scanned over a range of 11 GHz to cover one FSR of the FPIs and with that about five FSRs of the Fizeau interferometer. The data acquired during an ISR measurement contains 147 observations and each observation itself contains 3 different frequency steps which are spectrally separated by 25 MHz. Hence, the ISR frequency range is (3 × 147 − 1) × 25 MHz = 11 GHz. Each observation consists of 30 measurements, and each measurement consists of 20 laser pulses, whereas the data from the last laser pulse are not acquired during the measurement. Thus, an ISR 140 contains the data of 147 × (20 − 1) × (30) = 83790 laser pulses and each measurement at a certain frequency step consists of 10 measurements and contains the data of 190 laser pulses. These settings are not necessarily fixed but could be adapted if required.
The raw signal measured within this procedure undergoes several preprocessing steps before being used for further investigations. First, only the internal reference signal is extracted from the data product and analyzed for ISR mode measurements.

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It is worth adding here that the atmospheric signal is principally available. Internal reference acquisitions with a false pulse validity status or any other corrupt data are eliminated, but no such data was observed for the analyzed ISR data set presented here. The remaining Rayleigh channel signal is then corrected for the detection chain offset (DCO) by subtraction of the mean DCO level, which is implemented to avoid negative values in the digitization, as well as for the laser energy change occurring during the laser frequency scan (see also Fig. 2 and Fig. 3). Then, the Rayleigh signal is separated into the one originating form 150 the direct channel (ACCD pixel 1 to 8) and the one from the reflected channel (ACCD pixel 9 to 16). The output data for the Mie channel I outMie (f ) and the Rayleigh channel I outRay (f ) is given as the mean intensity per laser pulse according to where I total Mie/Ray (f ) is the total intensity detected per frequency step for the Mie channel or the Rayleigh channel, respectively, and N pulses = 190 is the number of laser pulses during one frequency step (10 measurements). Both quantities, N pulses 155 and I total Mie/Ray (f ) are reported per frequency step in a single so-called AUX-ISR auxiliary file for each ISR Aeolus data product.

Laser energy drift correction
As the UV output laser energy varies during the frequency scan that is executed during an ISR measurement, the intensity detected per frequency step I total Mie/Ray (f ) needs to be corrected accordingly. The trend of the transmitted laser energy is 160 monitored by a photo diode (PD-74) that is mounted in the respective laser transmitter UV section (FM-A and FM-B) behind a highly reflective mirror, an additional diffuser and neutral density filters used for further signal attenuation (Lux et al., 2020b). The mean laser energy versus laser frequency derived from the ISR measurements performed between October 2018 and March 2021 is shown in Fig. 2 for the FM-A period (left) and FM-B period (right), respectively. The laser frequency is referenced to the center frequency of the direct channel FPI transmission curve (see also  It can be seen that the laser energy changes considerably with frequency for both lasers FM-A and FM-B. For instance, at the beginning of FM-A operation (Fig.2, left, brownish colors), the laser energy was measured to be about 62.5 mJ at lower frequencies (≈ −2.5 GHz) and 53.0 mJ for higher frequencies (≈ 8.5 GHz), which corresponds to a signal decrease of about 170 15% during the frequency scan. Furthermore it is obvious that, for FM-A, the laser energy is largest for lower frequencies and decreases with increasing frequency. This is also true for the early FM-B phase until February/March 2020 when a change of the laser cold plate temperature (CPT) caused a spectral shift of the laser energy maximum to be closer to the FPI filter cross point where also the wind measurements are performed (≈ 2.8 GHz). The laser cold plate couples the laser with the laser radiator which in turn radiates the heat loss of the laser out to space. Additionally, it can be recognized that the overall 175 laser energy is decreasing throughout the operation time for both lasers, whereas the decrease rate is considerably larger for the FM-A period (Lux et al., 2020b). This circumstance is discussed in more detail in Sect. 5.1. In any case, it is obvious that the Mie and Rayleigh signals obtained during an ISR measurement need to be corrected for the varying laser energy. This is done in the Aeolus L1B processor according to

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where I out Mie/Ray (f ) is the DCO corrected raw data as given by Eq. (1) and E norm (f ) is the normalized mean laser pulse is the signal measured by PD-74 as shown in Fig. 2, and E PD−74 (f n=1 ) is the PD-74 measured energy at the first frequency step. Thus, E norm (f ) is not necessarily ranging from 0 to 1, as it is normalized arbitrarily to the value of the first data point.
A detailed analysis of the measured FPI transmission curves revealed that the energy correction of the short wave modula-185 tions works reasonably well, but the correction of the overall trend seems to be insufficient. This is especially obvious from the skewness that is visible in the relative residuals of the analyzed FPI transmission curves. Such a tilt is not explainable by incorrectnesses caused by the fit-model, as only symmetrical functions are used for the analysis (see also Sect. 4). Thus, it is likely that the energy drift detected by PD-74 is not completely representative of the internal Rayleigh channel signal. Hence, a modified normalized laser energy E normnew (f ) is needed for a proper energy correction. In particular, it turned out that an additional 190 linear correction according to E normnew (n) = E norm (n) + ξ × (n/440) is leading to satisfying results, with n = 1 to 441 being the number of data points available for ISR measurements and ξ is a correction factor that is derived by the analysis of FPI transmission curve residuals such that the residual exhibits no skewness anymore. This procedure is illustrated in Fig. 3 to the data reveals that the slope is close to zero, when E normnew (f ) is used for the laser energy correction (orange), whereas a significant skewness is obvious when E norm (f ) is used (blue). It can be seen that the relative deviations vary between −2% and 4% (peak-to-peak), whereas the distinct modulation is caused by an insufficient description of the spectral features of the 200 Fizeau reflection and modulations of the incident laser beam profile and/or the transmission over the Fizeau aperture (see also  (2) by using Enorm(f ) (blue dots) and Enorm new (f ) (orange dots) for the laser energy drift correction. To illustrate the small differences, the y-axes is plotted with logarithmic scale. (c): Relative residuals of the best fits according to Eq. (9) and line-fits.

Used datasets
The first ISR measurement in space was performed on 2 Sept. 2018, only 11 days after the satellite launch. The laser was operated at low laser pulse energies of about 11 mJ. On 8 Sept. 2018, the first ISR measurement at full laser pulse reported energy of about 64 mJ was performed to verify the co-registration of the spectrometers. Co-registration means the spectral alignment of the Mie USR center with the FPI filter cross point. After having changed the Rayleigh spectrometer cover temper-210 ature (RCT) and having adjusted the laser frequency accordingly, the first ISR with full laser energy (59 mJ) and co-registered spectrometers was performed on 10 October 2018. This is also the first ISR that is used in this study which ends with the ISR that was performed on 15 March 2021, the last measurement before ALADIN went to survival mode due to an instrument related anomaly. For the sake of completeness, the date, start time and mean laser energy of the ISR measurements analyzed in this study are summarized in table 2.

Analysis of ISR data
As explained in Sect. 3, ISR data yields the transmitted signal intensity through the Fizeau interferometer and the FPIs over a frequency range of 11 GHz. This data provides valuable information about the co-registration of the interferometers but also on the overall alignment conditions of the optical receiver as the spectral shape of the interferometer transmission curves depends on various parameters. Such parameters are the interferometer properties themselves (e.g. plate spacing, plate reflectivity, 220 index of refraction of the medium between the plates, plate surface quality), the spectral characteristics of the laser beam (e.g. diameter, divergence, intensity distribution) and the incidence angle of the laser beam onto the interferometers. Thus, the measurement of the interferometer transmission curves and the careful analysis with respective mathematical model functions allows to investigate potential changes and drifts of the aforementioned quantities. In the following, the model functions for analyzing the interferometer transmission curves are introduced for both the Rayleigh channel (double-edge FPIs) and the Mie 225 channel (Fizeau interferometer).

Fabry-Pérot interferometers
The particular characteristics of FPIs as well as the corresponding mathematical descriptions are comprehensively summarized in the textbooks by Vaughan (1989) and Hernandez (1986). Another illustrative mathematical description of the characteristics of an FPI that is applied in a direct detection wind lidar is given by McGill et al. (1997). In this section, the models used 230 to analyze the double-edge FPI transmission curves are demonstrated and corresponding parameters describing the overall alignment conditions of the ALADIN optical receiver are introduced. It will be shown that the sequential arrangement of the interferometers requires some special treatment. Parts of the model functions have already been developed before the launch of Aeolus based on particular measurements performed with the A2D (Witschas, 2011c;Witschas et al., 2012Witschas et al., , 2014 and were adapted to ALADIN.

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The transmission function T ideal (M ) of an ideal FPI (i.e., axially parallel beam of rays, mirrors perfectly parallel to each other, mirrors of infinite size, and mirrors without any defects) is described by the normalized Airy function according to where A accounts for any absorptive or scattering losses in or on the interferometer plates, R is the mean plate reflectivity, and M is the order of interference which can physically be considered as the number of half-waves between the interferometer 240 plates and which can be written as where f is the frequency of the transmitted light, n is the index of refraction of the medium between the plates, c is the velocity of light in vacuum, d is the plate separation and θ is the incidence angle of the illuminating beam. Furthermore, the frequency change that is needed to change M by one is defined as the FSR of the interferometer F FSR and is given by Additionally, the full width at half maximum ∆f FWHM of T ideal (f ) can be calculated according to where the approximation is valid if the argument of the inverse sine has small values, which is true in case of R being close to unity.

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In reality, however, imperfections and irregularities on the FPI mirror's surfaces cause a change in the intensity transmission of the FPI, which has to be considered when deriving appropriate model functions. Such deviations can for instance be caused by microscopic imperfections on the mirrors, errors in their parallel alignment, or non-uniformities in the reflective coatings which cause the effective mirror separation to vary across the face of the interferometer. As for instance shown by Vaughan (1989), different defect functions can be applied to the Airy function to deal with the various kinds of defects. In case of 255 ALADIN it turned out that a normally distributed Gaussian defect function according to (Witschas, 2011c) is well suited for that purpose. Here, σ g is the standard deviation of the Gaussian defect function and is called defect parameter.
The convolution of Eq.
(3) and Eq. (7) leads to a modified FPI transmission function normalized to unit area according to where f 0 denotes the center frequency. The effect of absorptive or scattering losses is neglected here. In case of ALADIN, also the sequential arrangement of the interferometers needs to be taken into account (see also Fig. 1), which on the one hand means that the photons within the receiver are recycled, but on the other hand means that any spectral imprint of the light reflected from one interferometer is also affecting the spectral characteristics of the transmitted light of the following interferometers.
Hence, for the direct channel FPI, the spectral characteristics of the light reflected from the Fizeau interferometer have to be 265 considered. Accordingly, for the reflected channel FPI the spectral characteristics of the light reflected from the direct channel FPI have to be considered. Thus, the spectral shape of the light transmitted through the direct channel FPI T dir (f ) is described where I dir is the mean intensity per FSR, and R Fiz (f ) depicts the reflection on the Fizeau interferometer which is described 270 by an empirically derived formula according to where I Fiz is the modulation depth (peak to peak), F FSRFiz is the FSR of the Fizeau interferometer, f 0Fiz is the center frequency (valley of the cosine function), and d Fiz is the y-axes shift from zero and is set to be constant (d Fiz = 0.5). Although Eq. (10) is only an approximation of the complex and varying reflection function of the Fizeau interferometer, it provides suf-275 ficient accuracy. This is demonstrated in Fig. 4, which shows the normalized Fizeau reflection depending on the commanded laser frequency obtained from the ISR measurement performed on 10 October 2018 (black dots) and the corresponding leastsquares best-fit using Eq (10) (light blue line). In particular, what is contained in the AUX-ISR auxiliary file is the signal transmitted through the Fizeau interferometer depending on the commanded laser frequency T Fiz (f ). Based on that, the reflected signal is calculated without considering any absorption or scattering losses with R Fiz (f ) = 1 − T Fiz (f ). The overall 280 spectral shape of the Fizeau reflection is well represented by the fit in spectral regions where measurement data is available. In regions where the Mie fringe is out of the USR and not imaged onto the ACCD (e.g., 2.5 GHz to 3.0 GHz), no comparison can be performed. To describe the transmission through the reflected channel FPI one additionally has to consider the reflection on the direct channel FPI and furthermore a potentially leaking beam splitter (see also PBSB in Fig. 1) that could partly lead to a direct 285 illumination of the reflected channel FPI. Considering that, the transmission through the reflected channel where T dir = T dir (f )/T dir (f 0 dir ) is the normalized transmission function of the direct channel FPI. Q takes into account a potentially leaking polarizing beam splitter, allowing for a T ref (f 0 dir ) different from zero, with zero being the value of the ideal 290 case. All other parameters are for the reflected channel as described for the direct channel in Eq. (9).
To investigate the ALADIN instrumental alignment and ongoing spectral drifts, a fit of Eq. (9) and Eq. (11)  Based on the determined fit parameters, further quantities that characterize the FPI transmission curves can be derived.
The FWHM of an ideal FPI was already introduced by Eq. (6). After introducing a defect parameter that takes into account 300 any imperfections and irregularities on the FPI mirror's surfaces, the FWHM can be calculated by describing the convolution of an Airy function and a Gaussian function by a Voigt function whose FWHM can be accurately approximated (Vaughan, 1989;Olivero and Longbothum, 1977). Without considering the reflection on the Fizeau interferometer, the total FWHM where ∆f FWHM ref is given by Eq. (6), and ∆f FWHM def = 2 √ 2 ln 2 σ g , which accounts for the broadening by defects. Thus, ∆f FWHMtot provides a good possibility to monitor the characteristic FPI transmissions without being influenced by the Fizeau interferometer.

Fizeau interferometer
In a Fizeau interferometer the plates are set with a wedge angle and spacing chosen to match the spectroscopic problem. The 310 resultant fringes are thus localized at the plates rather than at infinity as in the FPIs. Furthermore, the fringes are straight lines rather than circular, and aligned parallel to the wedge vertex. A text book analysis of the particular characteristics of Fizeau interferometer is given by Born and Wolf (1980), drawing on the analyses of Brossel (1947) and has since been extended by many authors (e.g., Kajava et al., 1994;McKay, 2002). These calculations all essentially use classical ray optic techniques and typically show asymmetric fringes often with appreciable fringe satellites, particularly for larger wedge angles. In this simple 315 ray optic treatment no allowance is made for the local slope of the plates and no account is taken of diffraction effects.
For the Aeolus Fizeau interferometer the wedge angle is rather small (4.77 µrad) and the plates themselves are subject to "fine grain" surface defects of regular spiral character, due to the magnetorheological optical finishing process. In this situation the resultant fringes can only be modelled by rigorous wave optic techniques (e.g., Jakeman and Ridley, 2006;Vaughan and Ridley, 2013). The resultant wave optic fringe profiles show some asymmetries, but under the Aeolus conditions of operation (small wedge angle and surface defects less than about ±1 nm) these were shown to be relatively small and the fringes could be sufficiently described by a Lorentzian function. Hence, for a respective ISR data set, the fringes originating at each frequency step are analyzed by fitting a Lorentzian curve according to where I Peak height is the peak amplitude, ∆f FWHMFiz the FWHM and x 0 the position of the fringe and which is usually called 325 Mie response. The pixels of the ACCD are numbered from 1 to 16. Thus, when the Fizeau fringe is centered on the ACCD, the center position is half way between pixel (px) 8 and 9, namely 8.5 px. Hence, each pixel index value from 1 to 16 denotes the center of each ACCD column, resulting in a start value of 0.5 px and an end value of 16.5 px. The fit itself is performed by applying a downhill simplex optimization procedure with Eq. (13). Compared to the FPI analysis, the analysis of the Fizeau fringes is already performed by the Aeolus L1B processor. Thus, I Peak height , ∆f FWHMFiz and the Mie response x 0 are available 330 in the AUX-ISR product files.
Furthermore, the Fizeau transmission is calculated at each frequency step as the intensity sum of all 16 pixel after DCO correction. Additionally, the Fizeau transmission is corrected for the laser energy change occurring during the frequency scan similar to the Rayleigh channel signals (see also Eq. (2)), however, an additional energy drift as described in Sect. 3.2 and as it is applied for the Rayleigh signal is not considered for the Mie signal. The evaluation of the Fizeau transmission, hence, gives 335 an approximation of potential changes in the beam intensity profile or beam diameter in one dimension.

Instrument functions for the Fizeau interferometer and the FPIs on 10 October 2018
The first ISR with full laser energy (59 mJ) and co-registered spectrometers was performed on 10 October 2018. The FPI transmission curves measured on that day including model fits according to Eq. (9) and Eq. (11), the corresponding relative residuals as well as the derived Mie response are shown in Fig. 5. The corresponding fit results are summarized in table 3.

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The given uncertainty of the fit values denotes the standard error derived by the fit routine and the last column indicates the specification values for comparison.
In the top panel (a), the measured FPI transmission curve of the direct channel is indicated by light blue circles and the one of the reflected channel by yellow circles given in least significant bits (LSB) which represent the digitized counts for the photon flux. The transmission curves were laser energy corrected by using E normnew (f ) (see also Eq. (2)). The corresponding 345 best-fits according to Eq. (9) and Eq. (11)  spacing on both sides of the filter curves is well within the specification of (5477 ± 120) MHz. a commanded frequency of about 1.75 GHz, which is also the frequency used for wind measurements (dashed magenta line).
In the middle panel of Fig. 5 (b), the relative residual of the direct and the reflected channel as well as linear-fits to the data are 365 depicted by the blue and orange line, respectively. The relative deviations vary between −2% and 4% (peak-to-peak), whereas the distinct modulation is caused by an insufficient description of the spectral features of the Fizeau reflection (Eq. (10)) and modulations of the incident laser beam profile and/or the transmission over the Fizeau aperture. However, as the Fizeau reflection cannot be measured directly with the instrument being in space, it is difficult to provide a better description without correcting other features that may have a different origin. Still, these features are only a few % in amplitude and very constant 370 over time. Thus, the other fit parameters and their temporal trends can be considered to be reliable and not impacted. It is also worth mentioning here, that the shown deviations cannot directly be related to a potential systematic error of the retrieved wind speeds, as several steps are performed during the wind processing chain. The only processing step that directly applies FPI transmission fit curves is the RBC that considers the impact of different atmospheric temperatures and pressures on the receiver response (Dabas et al., 2008;Dabas and Huber, 2017). Within the RBC, the FPI fit curves are convolved with 375 Rayleigh-Brillouin spectra of different temperatures and pressures, as well as with a tilted top-hat function to consider the different optical illumination between the internal reference path and the atmospheric path. The particular accuracy of the latter procedure cannot be well assessed, as there is no possibility to measure the FPI transmission curves via the atmospheric path with the needed accuracy. Among others, this is the reason why additional bias corrections based on ground return signals or ECMWF-model data are performed to obtain a wind product with a small systematic error of below e.g. ≈ 1 ms −1 . This 380 bias correction is extensively explained by Weiler et al. (2021b). Additionally, as the width (FWHM) of the Rayleigh-Brillouin spectrum is rather broad (i.e. 3 GHz to 4 GHz for a laser wavelength of 354.8 nm and atmospheric temperatures and pressures), the response of the Rayleigh channel is insensitive to small scale details as observed for the FPI transmission curve residuals.
As it was mentioned in section 3.2, the laser energy drift correction may have an impact on the derived FPI transmission curves. Thus, for the sake of completeness, the fit parameters shown in table 3 are given for both, data that was corrected with 385 E normnew (f ), as well as for data that was corrected with E norm (f ) given in brackets (see also Sect. 3.2). It can be seen that the energy drift correction has only a minor impact on the retrieved fit parameters which lies within the fit error. Only the obtained mean intensity per FSR shows differences of about 1% for the direct channel and 3% for the reflected channel data.
n/a = no specification available. about ±0.6 GHz around each USR center frequency or e.g. the FPI filter cross point. The Mie USR is usually projected onto the range between pixel column 4 to 14 of the ACCD. The wind measurement is performed at a commanded frequency of 1.75 GHz, resulting in a Mie fringe being located almost in the center of the ACCD detector (pixel 9.2).
As these kind of ISR measurements have been performed on a regular weekly basis, they offer the opportunity to analyze 395 time series of the discussed fit parameters and with that to investigate spectral drifts and a change of the alignment conditions of the instrument as discussed in Sect. 5.

Temporal evolution of the Fizeau and FPI spectral transmission curves
In this section, respective fit parameters and their temporal evolution are discussed. These are the detected mean intensity per FSR of the FPIs (Sect. 5.1), the FPI center frequencies and the corresponding spectral spacing (Sect. 5.2), the FPI To first demonstrate the notable alignment stability of the Aeolus optical receiver as well as the reproducibility of ISR 405 measurements, all normalized FPI transmission curves are shown in Fig. 6 for the FM-A period (a) and the FM-B period (b).
To be able to directly compare respective measurements, the direct channel transmission curves are normalized to unit area such that FFSR 0 T dir (f ) df = 1, and the reflected channel is normalized accordingly with the same factor as used for the direct channel to keep the ratio between the two respective channels. Furthermore, the x-axes is normalized to the direct channel center frequency and thus marks the 0 GHz. Brownish colors correspond to the early time of the respective laser period, and 410 blueish colors to the later times (see also the dates shown in the label of each panel). For the FM-A period, the direct channel transmission curve is very reproducible. No distinct changes can be recognized except for slight changes originating at a normalized frequency of about −2 GHz which might be caused by a spectral change of the Fizeau reflection. This is different for the reflected channel transmission curves. Here, a remarkable drift of both, the center frequency as well as the peak intensity can be observed. As the curves are normalized, this drift is with respect to 415 the direct channel in both x-direction (frequency) and y-direction (intensity). Additionally, spectral changes are obvious and probably also caused by changing spectral characteristics of the Fizeau reflection.
For the FM-B period, a center frequency drift between the respective channels is much less or rather not observable. This will also be discussed in more detail in Sect. 5.2, which deals with the accurate analysis of the time series of the FPI center frequencies. As for the FM-A period, the peak intensity ratio varies with time. Additionally, it can be recognized that the noise 420 on the transmission curves starts to increase remarkably around August 2020 which is caused by enhanced signal modulations on the internal reference path.

Mean intensity
One quantity that can directly be obtained from the FPI transmission curve analysis by means of Eq. (9) or rather Eq. (11) is 15% (i.e. from 4380 LSB to 3740 LSB for the direct channel), but also to a remarkable reduction of the decrease rate namely to (−1.9±0.1) LSB/d for the direct channel and (−1.5±0.2) LSB/d for the reflected channel. Interestingly, the mean laser pulse energy (Fig. 7, bottom) increased after the CPT changes indicating that these changes may have led to alignment changes that induced differences in the signal levels on the internal path including the Rayleigh ACCD and the PD-74. Since August 2020 the determined fit results from week to week got more variable due to appearing signal fluctuations in the internal reference 465 signal. At the beginning of December 2020, a laser energy increase of about 15% was obtained by changing laser operating parameters, however, since mid December 2020 the decrease rate increased to be (−11.4 ± 0.4) LSB/d for the direct channel and (−9.9 ± 0.3) LSB/d for the reflected channel. After the laser parameter changes in December 2020, also the intensity ratio changed rapidly (within 4 weeks) from about 0.78 to 0.83, which is another hint that the instrumental alignment changed significantly within this time period. Further indications for ongoing alignment changes can be derived from the time series of 470 the FPI center frequencies as discussed in the next section.

FPI center frequencies and spectral spacing
The center frequencies derived by fitting Eq. (9) and Eq. (11)  towards the same spectral direction (towards higher frequencies) with a comparable rate. For period 3, the center frequency drift was (0.22 ± 0.01) MHz/d (direct channel) and (0.16 ± 0.01) MHz/d (reflected channel), whereas for period 4 it was (0.14 ± 0.01) MHz/d (direct channel) and (0.16 ± 0.01) MHz/d (reflected channel). Thus, the drift rate decreased for the direct channel but stayed constant for the reflected channel. This indicates that the overall alignment conditions or in particular the initial incidence angles for the internal reference path were remarkably different for the different lasers FM-A and FM-B.

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In Sect. 6, these observations are used to further estimate the underlying reason for these spectral drifts.
Another quantity that can be derived from the center frequencies of both Rayleigh channels is the spectral spacing as shown in the bottom panel of Fig. 8 (black circles). The spacing is an important measure as an unconsidered spacing drift would lead to systematic errors in the retrieved wind speeds, whereas an equal center frequency drift with similar rate and spectral drift direction (e.g. as it is almost true for the FM-B period) would not affect the wind retrieval. At the beginning of the mission, the 515 spacing is determined to be 5450 MHz and then decreased rapidly to smaller values, which is a result of the center frequency drift occurring towards different spectral directions. Still, the drift of the spacing shows a settlement which is even independent of the switch to the FM-B laser in June 2019. Though the overall spacing changes by about 30 MHz due to the different illumination conditions with the different lasers, the overall settlement of the drift continues. This is also confirmed by an exponential fit (Fig. 8, bottom, gray line)) applied to the FM-A data set that indicates an asymptotic convergence at 5320 MHz, 520 which is about the spacing determined for FM-B in early 2021. This finding would allow to expect the drifting optical element being located not in the laser transmitters but between laser transmitter and optical receiver bench or for an optical element, which is common to the internal path of FM-A and FM-B if not even a rigid body motion of the laser and/or receiver optical benches with respect to each other.
Furthermore, during the FM-B period, the spacing shows some distinct drift periods as for instance around December 2019 525 and around June 2020. Comparing these drifts with the ALADIN ambient temperature measured at the DEU of the system (Fig. 8, bottom, magenta line) a correlation (June 2020) and anti-correlation (December 2019 and December 2020) is obvious, whereas the temperature changes are caused by entering/leaving a solar eclipse phase where parts of the satellite orbit are in darkness leading to a temperature decrease of the instrument. This finding confirms that the ambient temperature in the system changes and that these temperature changes have an impact on the overall alignment of the instrument, even though the 530 temperature of the FPIs only changes by 10 mK during these eclipse periods (not shown).

Full width at half maximum
The FWHM of the FPI transmission curves can be calculated according to Eq. (12) using the plate reflectivity and the defect parameter determined by the fit of Eq. (9) and Eq. (11) to the measured FPI transmission curves. The FWHM for the ISR data sets obtained from 10 October 2018 until 15 March 2021 (see also table 2) are shown in Fig. 9 for the direct channel (blue 535 circles) and the reflected channel (orange circles), respectively.
It can be seen that the FWHM of both channels was rather similar at the beginning of the mission namely about 1590 MHz.
During the FM-A period, the direct channel

Fizeau reflection spectral position
The reflection on the Fizeau interferometer has an impact on the FPI transmission curves. Hence, drifts of the spectral characteristics of the Fizeau interferometer reflection can also be derived from FPI analyses. As shown with Eq. (10), the center frequency f 0Fiz of the Fizeau reflection is a free fit parameter for the model functions describing the FPI transmission curves. Fizeau center frequencies settled until about August 2020. Since then, the obtained fit parameters are in general more variable due to a larger variability of the spectrometer signals which could be explained by beam clipping happening due to the ongoing alignment drift (see also Fig. 7 and Fig. 8). Besides the analyses of the Fizeau reflection impact on the Rayleigh channel signals, also Mie channel signals are available from ISR measurements for further investigations of potentially ongoing spectral drifts.
The results of the Mie signal analyses are discussed in the next section.

Fizeau intensity
The Fizeau intensity is calculated as the sum of the laser energy and DCO corrected Mie signal at each frequency step, and thus, gives an approximation of potential changes in the beam intensity profile or beam diameter that is illuminating the Fizeau interferometer (in one dimension). It is worth mentioning that any changes of the measured Fizeau intensity are mainly caused by a variation of the interferometer illumination of the internal path rather than due to a change of the Fizeau transmission itself,  What immediately can be seen is that the Fizeau transmission looks different for FM-A and FM-B, and that it evolves with time. Although the pronounced maximum around the Mie response of 9 px is similar for FM-A and FM-B, the distribution for smaller Mie responses looks different, which points to a different intensity distribution of the illuminating beam or rather different illumination conditions as for instance clipping on other optical elements. Additionally, it can be recognized that for both lasers, the width of the Fizeau transmission is decreasing with time which could be explained by a shrinking beam 585 diameter or a change in the divergence of the illuminating beam. Furthermore, it is obvious that not only the width but also the overall spectral features are evolving which might be explained by a changing intensity distribution of the illuminating beam.
6 Discussion of spectral drifts In Fig. 8 of Sect. 5.2, the time series of the determined FPI center frequencies was shown and demonstrated different drift behavior for the respective lasers FM-A and FM-B. To understand the observed results and to relate them to respective align-ment changes, the equations discussed in Sect. 4.1 have to be revised as they only consider an ideal FPI with mirrors of infinite size, being illuminated normal to the optical axis of the FPI plates by a perfectly collimated beam. In reality however, the illumination cone of the light beam passing through the FPI has a certain FOV with an angular radius θ F , also called the input divergence (half of the full cone angle). For the ALADIN internal reference signal θ FINT is assumed to be 455/2 µrad, as determined by the parameters of the reference laser beam as it enters the FPI. In contrast, for the atmospheric signal, θ FATM 595 is estimated to be 1.44/2 mrad, as determined by a pinhole aperture in the optical chain. Furthermore, the light beam may illuminate the FPI at a certain angle of incidence θ A . Such a situation is illustrated in Fig. 12 for θ A ≈ 2 · θ F with 0 denoting the prime optical axis of the system defined as the normal to the FPI plates. Hence, the FPI circular interference fringes (black rings, not shown for the full circle) are centered at 0, but for an off-axis aperture (gray circle) centered at θ A only portions of the fringes will be illuminated as indicated by the orange, light-blue and dark-blue circular arcs. The dashed lines exemplarily 600 mark the crossing points of the second interference fringe with the aperture or rather the fraction of the second interference fringe that is transmitted through the aperture (light blue circular arc). The asymmetric behavior of such an off-axis illuminated FPI has been treated in detail by Hernandez (1974) as well as in the text books by Hernandez (1986) and Vaughan (1989), whereas only the most important points are recapitulated here. The following analysis is based on the principle demonstrated in Fig. 12.

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The ideal FPI, as discussed in Sect. 4.1, is an angle dependent filter with quadratic dispersion. In particular, the transmission for a narrow beam (θ F → 0) at a certain θ A experiences a frequency shift ∆f compared to the beam at normal incidence where θ 1 = λ/d = 5.093 mrad is the angular radius of the first full fringe and is used as a scaling reference (Vaughan, 1989) 610 as is relates certain angular radii together with Γ FSR to a frequency shift compared to the beam at normal incidence. More precisely, θ 1 is the angular radius of the first fringe from the center supposing that a bright fringe of zeroth order exists exactly at the center of the pattern. Hence, the radius of the first full fringe corresponds to one FSR. From Eq. (14) it can also be seen that any change of the angle of incidence would lead to a center frequency increase as it is true for the FM-B period. For ALADIN, λ = 354.8 nm and d = 13.68 mm and Γ FSR = 10.95 GHz are given according to table 1. Additionally, by replacing 615 θ 1 and Γ FSR with their definitions given above and in Eq. (5), the relative frequency shift (∆f /f ) compared to the beam at normal incidence is derived to be Figure 12. Illustration of FPI operation with an angle of incidence θA and a field of view θF in angular space with 0 denoting the prime optical axis of the system defined as the normal to the FPI plates. The FPI circular interference fringes (black rings) are centered at 0, but only portions of the fringes will be illuminated due to the off-axis centered aperture (gray circle) as indicated by the orange, light-blue and dark-blue circular arcs. The dashed lines exemplarily mark the crossing points of the second interference fringe with the aperture or rather the fraction of the second interference fringe that is transmitted through the aperture (light blue circular arc). The detailed analysis leading to Eqs. (14) to (19) and to the dispersion curves shown in Fig. 13 are based on the principle demonstrated here as it is also applicable for the dependence of the zero-order spot imaged on the Aeolus ACCD.
For a larger beam with a non-negligible θ F , two regions of operation may be considered with θ A θ F and θ A > θ F . For θ F > 0 and nominal incidence (θ A = 0), the aperture profile extends out to and is a top-hat function with a full width ∆f 0 . The median position of this aperture function, which denotes the center of energy, is at half of this value i.e. ∆f M = ∆f 0 /2 and the peak intensity is usually designated as I 0 . For 0 < θ A θ F , the aperture profile starts to become asymmetric and extends out to a full width ∆f W given by 625 however, the median position of the energy distribution remains constant up to θ A ≈ 0.293 · θ F according to ∆f M = ∆f 0 /2.
For 0.293 · θ F < θ A θ F , the aperture profile becomes increasingly asymmetric with a full width given by Eq. (17). However, the peak intensity remains at I 0 . The median position θ M is given by where κ ≈ 0.6 to a good approximation. When θ A > θ F , the incident beam no longer overlaps the optical axis normal to the 630 FPI plates and the angular position of the peak of the profile is at θ p given by θ p 2 = θ A 2 − θ F 2 and in frequency terms and the peak intensity I p is given by However the median value which denotes the energy center of the profile is still calculated by Eq. (18) with κ ≈ 0.6.

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The corresponding dispersion curves of frequency shift (i.e. median position) versus angle of incidence θ A , calculated by means of Eq. (14) to Eq. (20) for beams of different angular aperture θ F are shown in Fig. 13. The green line denotes the ideal case with θ F = 0, the red line indicates the case for the internal reference beam with θ FINT = 455/2 µrad, and the black line depicts the case for the atmospheric signal beam with θ FATM = 1.44/2 mrad.
With Fig. 13 it can be seen that incidence angle drifts of a few hundred µrad are needed to explain the observed center 640 frequency drifts (see also table 4) for the internal reference beam. It is further obvious that the exact incidence angle drift is depending on the initial incidence angle which is essentially unknown. During the FM-A period, the center frequency drifts by +39 MHz for the direct channel and by −75 MHz for the reflected channel. Thus, as the center frequency decreases for the reflected channel, it can be concluded that the initial incidence angle for the reflected channel was definitely different from normal incidence. For instance, an initial incidence angle of about θ Ai = 425 µrad and a drift towards normal incidence could 645 explain the observed center frequency drift. For the direct channel, the observed center frequency drift of +39 MHz could be explained by an incidence angle change of about 325 µrad supposing θ Ai = 0. For θ Ai = 400 µrad, the incidence angle only has to drift by about 110 µrad to θ A = 510 µrad to explain the observed center frequency drifts.
For the FM-B period, the frequency drifts of both channels are comparable and with similar sign. In particular, the center frequency drifts by +96 MHz for the direct channel and by +87 MHz for the reflected channel. Supposing an initial incidence an- I d e a l I N T A T M Figure 13. Dispersion curves of frequency shift versus angle of incidence for beams of different angular aperture (FOV) with θF = 0 as corresponding to the ideal case (green), θF INT = 455/2 µrad as corresponding to the internal reference beam (red) and θF ATM = 1.44/2 mrad as corresponding to the atmospheric signal beam (black). The frequency shift between internal reference signal and atmospheric signal at nominal incidence is close to 100 MHz.
gle of θ Ai = 0 an incidence angle change of about 475 µrad is needed to explain the observed center frequency drifts. For larger initial incidence angles, the corresponding drift would accordingly be smaller. What can also be recognized during the FM-B period is that the center frequency drift rate for the direct channel is decreasing over time from (+0.22 ± 0.01) MHz/d (period 3) to (+0.14 ± 0.01) MHz/d (period 4). As the dispersion curve gets steeper for larger θ A , this behavior can only be explained by a decreasing θ A -drift rate or a clipping of the beam which would alter the apparent θ F .

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Additionally, from Fig. 13 it can be recognized that even for normal incidence (θ A = 0), the atmospheric signal is already shifted by about 100 MHz with respect to the internal reference signal, which is due to the considerably larger FOV of the atmospheric signal and which is an important difference between the two signal channels.
Summarized, it can be said that the overall alignment conditions were significantly different for the FM-A and FM-B period.
Furthermore, it has to be pointed out that a θ A -drift of several hundred µrad is rather significant and could well lead to clipping 660 of the beam, considering the angular size of the field stop at Rayleigh spectrometer level of 1440 µrad and an angular size of the Rayleigh spots of about 968 µrad (4σ − diameter). Thus, the larger variability of the retrieved fit parameters originating in August 2020 could be explained by larger variations induced by beam clipping. The spacing drift (Fig. 8, b) and the clear correlation/anti-correlation to the instrument temperature confirms the temperature sensitivity of the instrumental alignment Table 4. Frequency drift rates and total frequency drift obtained for the direct channel and the reflected channel over two periods of operation for FM-A and FM-B as shown in Fig. 8 to 10 mK even during the eclipse phases.

Summary
In August 2018, ESA has launched the first Doppler wind lidar into space. To calibrate the instrument and to monitor the overall instrument conditions, instrument spectral registration measurements have been performed with Aeolus on a weekly basis.
During these measurements, the laser frequency is scanned over a range of 11 GHz to measure the transmission curves of the 670 spectrometers. Within this study, tools and mathematical model functions to analyze the measured spectrometer transmission curves were introduced and used to retrieve time series of respective fit parameters for the time period from October 2018 to March 2021. The models representing the FPI transmission curves is based on an Airy function with Gaussian defects, but does also consider the spectral modification induced by the reflection on the Fizeau interferometer which is due to the sequential spectrometer setup. Additionally, the impact of the finite aperture within the receiver and the field of view of the illuminating 675 beam on the FPI transmission is discussed. Based on this analysis, it is revealed that the overall conditions were different for the respective lasers FM-A (August 2018 till June 2019) and FM-B (July 2019 till now).
The Rayleigh channel signal levels are shown to be remarkably smaller for the FM-A period. In particular, at the begin of the respective laser period, the FM-A signal levels are smaller by about 30%. Furthermore it is shown that the detected signal levels decreased over time. For the FM-A period, the decrease is rather constant for both channels (direct channel: −0.14 %/day, 680 reflected channel:−0.13 %/day) except for certain time periods that were related to laser parameter optimizations. As for FM-A, the signal levels are decreasing during the FM-B time period. A laser cold plate temperature optimization performed in March 2020 led to a signal decrease by 15% but also to a significant reduction of the decrease rate. Since December 2020, the signal levels are shown to decrease stronger than before.
The FPI center frequencies are shown to drift by several MHz per week throughout the mission. It is demonstrated that 685 during the FM-A period, the drift rates were rather variable and in different spectral directions for the respective FPI channel.
On the other hand, the drift was rather constant and towards similar spectral directions for both channels during the FM-B period. This indicates that the overall illumination or rather alignment conditions were rather different for the two lasers FM-A and FM-B. By considering the field of view of the internal reference beam (455 µrad) as well as the finite aperture size of the FPIs (1.44 mrad) it is shown that the incidence angle has to change by several hundred µrad to explain the observed center 690 frequency drifts. Considering the angular size of the field stop at Rayleigh spectrometer level (1.44 mrad) and an angular size of the Rayleigh spots (0.968 mrad, 4σ diameter) such a drift is quite significant and may lead to a clipping of the beam. It is further shown that changes of the ambient temperature affect the overall instrument stability, whereas the reflected channel is more sensitive than the direct channel. This is especially obvious in particular eclipse phases, where the satellite is partly out of sun illumination and thus decreases its temperature. The significant observed frequency drift also explains why regular 695 instrument calibrations are inevitable to avoid systematic errors in the Aeolus wind product.
In addition to the FPI transmission curves, the characteristics of the Fizeau transmission are analyzed directly as well as from the imprint on the FPI transmission curves. It is shown that the spectral shape of the Fizeau transmission is different for the FM-A and FM-B laser, and that it evolves with time. Furthermore, for both lasers the Fizeau transmission is decreasing with time which could be explained by a shrinking beam diameter or a reduction of the beam divergence.

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In the future is is foreseen to extend the presented results on Aeolus alignment drifts by using for instance the spatial information of the Rayleigh spots.
Furthermore it is pointed out that the instrumental functions and analysis tools introduced in the study may also be applied for upcoming missions using similar spectrometers as for instance EarthCARE (ESA) which is based on the Aeolus FPI design.
Author contributions. BW prepared the main part of the manuscript and performed the ISR analyses, CL contributed with the analysis of the 705 ALADIN laser performance, OL provided useful information on the ALADIN laser performance, corresponding time series and laser setting changes, UM performed processor modifications and provided special data sets, OR led the presented study and helped to prepare the paper manuscript, FW provided particular data sets for the presented study, FF developed FPI mathematical models to determine the ALADIN alignment conditions based on ISR measurements, TF and AD contributed with discussions and helped to prepare the paper manuscript, DH developed the operational Aeolus L1B processor and provided continuous support with special processing requests, MV performed 710 investigations regarding both the FPI and Fizeau interferometer alignment conditions and the resulting performance for Aeolus.
Competing interests. The authors declare that they have no conflict of interest.