3.1 Measurement procedure

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 contain 147 observations, and each observation itself contains 3 different frequency steps, which are spectrally separated by 25 MHz. Hence, the ISR frequency range is ($\mathrm{3}\times \mathrm{147}-\mathrm{1})\times \mathrm{25}\mathrm{MHz}=\mathrm{11}$ 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 contains the data of $\mathrm{147}\times (\mathrm{20}-\mathrm{1})\times \left(\mathrm{30}\right)$ = 83 790 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.

Figure 2Mean laser pulse energy versus normalized laser frequency derived from ISR measurements for the FM-A period (a) and FM-B period (b).

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Figure 3(a) Normalized laser energy E_norm(f) (blue) and corrected normalized energy $E_{{\text{norm}}_{\text{new}}}\left(f\right)$ (orange) versus commanded laser frequency for the ISR measurement performed on 10 October 2018 using ξ = 0.04. (b) Corresponding FPI transmission curves of the direct channel according to Eq. (2) by using E_norm(f) (blue dots) and $E_{{\text{norm}}_{\text{new}}}\left(f\right)$ (orange dots) for the laser energy drift correction. To illustrate the small differences, the y axis is plotted with logarithmic scale. (c) Relative residuals of the best fits according to Eq. (9) and line fits.

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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. 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 were 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 Figs. 2 and 3). Then, the Rayleigh signal is separated into the one originating form 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 $I_{{\text{out}}_{\text{Mie}}}\left(f\right)$ and the Rayleigh channel $I_{{\text{out}}_{\text{Ray}}}\left(f\right)$ are given as the mean intensity per laser pulse according to

$$\begin{array}{}\text{(1)}& I_{{\text{out}}_{\text{Mie/Ray}}}\left(f\right)={\displaystyle \frac{I_{{\text{total}}_{\text{Mie/Ray}}}\left(f\right)}{N_{\text{pulses}}}},\end{array}$$

where $I_{{\text{total}}_{\text{Mie/Ray}}}\left(f\right)$ 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 and $I_{{\text{total}}_{\text{Mie/Ray}}}\left(f\right)$, are reported per frequency step in a single so-called AUX-ISR auxiliary file for each ISR Aeolus data product.

3.2 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 $I_{{\text{total}}_{\text{Mie/Ray}}}\left(f\right)$ needs to be corrected accordingly. The trend of the transmitted laser energy is 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 Fig. 8). Thus, 0 GHz indicates the center frequency of the direct-channel FPI. Brownish colors correspond to the early time of the respective laser period and blueish colors to the later times (see also the label of each panel).

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. 2a, 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 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 in the laser cold plate temperature (CPT) caused a spectral shift in the laser energy maximum to be closer to the FPI filter crossing 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 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

$$\begin{array}{}\text{(2)}& I_{{\text{total}}_{\text{Mie/Ray}}}\left(f\right)={\displaystyle \frac{I_{{\text{out}}_{\text{Mie/Ray}}}\left(f\right)}{E_{\text{norm}}\left(f\right)}},\end{array}$$

where $I_{{\text{out}}_{\text{Mie/Ray}}}\left(f\right)$ is the DCO-corrected raw data as given by Eq. (1), and E_norm(f) is the normalized mean laser pulse energy calculated according to E_norm(f) = $\left(E_{\text{PD-74}}\right(f\left)\right)/\left(E_{\text{PD-74}}\right(f_{n=\mathrm{1}}\left)\right)$, where E_PD-74(f) 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) does not necessarily range 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 modulations 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 incorrectness 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 $E_{{\text{norm}}_{\text{new}}}\left(f\right)$ is needed for a proper energy correction. In particular, it turned out that an additional linear correction according to $E_{{\text{norm}}_{\text{new}}}\left(n\right)$ = $E_{\text{norm}}\left(n\right)+\mathit{\xi }\times (n/\mathrm{440})$ leads 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, where Fig. 3a shows the laser energy measured by PD-74 and normalized to the first data point E_norm(f) (blue) as well as the corrected normalized laser energy $E_{{\text{norm}}_{\text{new}}}\left(f\right)$ (orange) for the ISR measurement performed on 10 October 2018. In Fig. 3b, the respective FPI transmission curves of the direct channel, derived by using E_norm(f) (blue) and $E_{{\text{norm}}_{\text{new}}}\left(f\right)$ (orange) for the laser energy drift correction, are shown. The corresponding relative residuals are shown in Fig. 3c. The line fits applied to the data reveal that the slope is close to zero when $E_{{\text{norm}}_{\text{new}}}\left(f\right)$ 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 Fizeau reflection and modulations of the incident laser beam profile and/or the transmission over the Fizeau aperture (see also Fig. 5 and the corresponding discussion). For the ISR measurement on 10 October 2018, ξ was determined to be 0.04. For the entire mission time discussed in this paper (October 2018 until March 2021), ξ varies between 0.05 and −0.18. At the beginning of the mission, this energy correction had to be performed manually; however, since December 2018, with the implementation of the L1B processor version 7.05, the additional energy drift correction was added.

Table 2Overview of the ISR data set used in this study.

^* Mean laser energy and standard deviation per ISR measurement.

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3.3 Used data sets

The first ISR measurement in space was performed on 2 September 2018, only 11 d after the satellite launch. The laser was operated at low laser pulse energies of about 11 mJ. On 8 September 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 crossing point. After having changed the Rayleigh spectrometer cover temperature (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.

4.1 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 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 is 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., 2012, 2014) and were adapted to ALADIN.

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

$$\begin{array}{}\text{(3)}& \begin{array}{rl}{\mathcal{T}}_{\text{ideal}}\left(M\right)=& {\left(\mathrm{1}-{\displaystyle \frac{\mathcal{A}}{\mathrm{1}-R}}\right)}^{\mathrm{2}}\left({\displaystyle \frac{\mathrm{1}-R}{\mathrm{1}+R}}\right)\\ & \times \left(\mathrm{1}+\mathrm{2}\sum _{k=\mathrm{1}}^{\mathrm{\infty }}{R}^k\cos \left(\mathrm{2}\mathit{\pi }kM\right)\right),\end{array}\end{array}$$

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 to be the number of half waves between the interferometer plates and which can be written as

$$\begin{array}{}\text{(4)}& M={\displaystyle \frac{\mathrm{2}n}{c}}df\cos \left(\mathit{\theta }\right),\end{array}$$

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 a 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 1 is defined as the FSR of the interferometer ℱ_FSR and is given by

$$\begin{array}{}\text{(5)}& {\mathcal{F}}_{\text{FSR}}={\displaystyle \frac{c}{\mathrm{2}nd\cos \left(\mathit{\theta }\right)}}.\end{array}$$

Additionally, the full width at half maximum Δf_FWHM of 𝒯_ideal(f) can be calculated according to

$$\begin{array}{}\text{(6)}& \begin{array}{rl}\mathrm{\Delta }{f}_{\text{FWHM}}& ={\mathcal{F}}_{\text{FSR}}\cdot \arcsin \left((\mathrm{1}-R)/(\mathit{\pi }\cdot \sqrt{R})\right)\\ & \approx {\mathcal{F}}_{\text{FSR}}\cdot \left((\mathrm{1}-R)/(\mathit{\pi }\cdot \sqrt{R})\right),\end{array}\end{array}$$

where the approximation is valid if the argument of the inverse sine has small values, which is true in the case of R being close to unity.

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 the case of ALADIN it turned out that a normally distributed Gaussian defect function according to (Witschas, 2011c)

$$\begin{array}{}\text{(7)}& {\mathcal{D}}_g\left(f\right)={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathit{\pi }}{\mathit{\sigma }}_{\mathrm{g}}}}\exp \left({\displaystyle \frac{f^{\mathrm{2}}}{\mathrm{2}{{\mathit{\sigma }}_{\mathrm{g}}}^{\mathrm{2}}}}\right)\end{array}$$

is well suited for that purpose. Here, σ_g is the standard deviation of the Gaussian defect function and is called the defect parameter. The convolution of Eqs. (3) and (7) leads to a modified FPI transmission function normalized to unit area according to

$$\begin{array}{}\text{(8)}& \begin{array}{rl}{\mathcal{T}}_{\text{Gauss}}\left(f\right)=& {\displaystyle \frac{\mathrm{1}}{{\mathcal{F}}_{\text{FSR}}}}\left(\mathrm{1}+\mathrm{2}\sum _{k=\mathrm{1}}^{\mathrm{\infty }}{R}^k\cos \left({\displaystyle \frac{\mathrm{2}\mathit{\pi }k}{{\mathcal{F}}_{\text{FSR}}}}(f-f_{\mathrm{0}})\right)\right.\\ & \left.\times \exp \left(-\mathrm{2}{\left({\displaystyle \frac{\mathit{\pi }k{\mathit{\sigma }}_g}{{\mathcal{F}}_{\text{FSR}}}}\right)}^{\mathrm{2}}\right)\right),\end{array}\end{array}$$

where f₀ denotes the center frequency. The effect of absorptive or scattering losses is neglected here. In the case of ALADIN, the sequential arrangement of the interferometers also 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 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 according to

$$\begin{array}{}\text{(9)}& \begin{array}{rl}{\mathcal{T}}_{\text{dir}}\left(f\right)=& {\mathcal{I}}_{\text{dir}}\cdot \left(\mathrm{1}+\mathrm{2}\sum _{k=\mathrm{1}}^{\mathrm{\infty }}{R_{\text{dir}}}^k\cos \left({\displaystyle \frac{\mathrm{2}\mathit{\pi }k}{{\mathcal{F}}_{{\text{FSR}}_{\text{dir}}}}}(f-f_{{\mathrm{0}}_{\text{dir}}})\right)\right.\\ & \left.\times \exp \left(-\mathrm{2}{\left({\displaystyle \frac{\mathit{\pi }k{\mathit{\sigma }}_{g_{\text{dir}}}}{{\mathcal{F}}_{{\text{FSR}}_{\text{dir}}}}}\right)}^{\mathrm{2}}\right)\right)\cdot {\mathcal{R}}_{\text{Fiz}}\left(f\right),\end{array}\end{array}$$

where ℐ_dir is the mean intensity per FSR, and ℛ_Fiz(f) depicts the reflection on the Fizeau interferometer, which is described by an empirically derived formula according to

$$\begin{array}{}\text{(10)}& {\mathcal{R}}_{\text{Fiz}}\left(f\right)=\left(\mathrm{1}-{\mathcal{I}}_{\text{Fiz}}\left(\cos {\left({\displaystyle \frac{\mathit{\pi }}{{\mathcal{F}}_{{\text{FSR}}_{\text{Fiz}}}}}(f-f_{{\mathrm{0}}_{\text{Fiz}}})\right)}^{\mathrm{4}}-d_{\text{Fiz}}\right)\right),\end{array}$$

where ℐ_Fiz is the modulation depth (peak to peak), ${\mathcal{F}}_{{\text{FSR}}_{\text{Fiz}}}$ is the FSR of the Fizeau interferometer, $f_{{\mathrm{0}}_{\text{Fiz}}}$ is the center frequency (valley of the cosine function), and d_Fiz is the y-axis 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 sufficient 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 least-squares 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 spectral shape of the Fizeau reflection is well represented by the fit in spectral regions where measurement data are available. In regions where the Mie fringe is out of the USR and not imaged onto the ACCD (e.g., 2.5 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 illumination of the reflected-channel FPI. Considering that, the transmission through the reflected channel 𝒯_ref(f) is described according to

$$\begin{array}{}\text{(11)}& \begin{array}{rl}{\mathcal{T}}_{\text{ref}}\left(f\right)=& {\mathcal{I}}_{\text{ref}}\cdot \left(\mathrm{1}-Q\stackrel{\mathrm{‾}}{{\mathcal{T}}_{\text{dir}}}\right)\\ & \times \left(\mathrm{1}+\mathrm{2}\sum _{k=\mathrm{1}}^{\mathrm{\infty }}{R_{\text{ref}}}^k\cos \left({\displaystyle \frac{\mathrm{2}\mathit{\pi }k}{{\mathrm{\Gamma }}_{{\text{FSR}}_{\text{ref}}}}}(f-f_{{\mathrm{0}}_{\text{ref}}})\right)\right.\\ & \left.\times \exp \left(-\mathrm{2}{\left({\displaystyle \frac{\mathit{\pi }k{\mathit{\sigma }}_{g_{\text{ref}}}}{{\mathrm{\Gamma }}_{{\text{FSR}}_{\text{ref}}}}}\right)}^{\mathrm{2}}\right)\right)\cdot {\mathcal{R}}_{\text{Fiz}}\left(f\right),\end{array}\end{array}$$

where $\stackrel{\mathrm{‾}}{{\mathcal{T}}_{\text{dir}}}={\mathcal{T}}_{\text{dir}}\left(f\right)/{\mathcal{T}}_{\text{dir}}\left(f_{{\mathrm{0}}_{\text{dir}}}\right)$ is the normalized transmission function of the direct-channel FPI. Q takes into account a potentially leaking polarizing beam splitter, allowing for a ${\mathcal{T}}_{\text{ref}}\left(f_{{\mathrm{0}}_{\text{dir}}}\right)$ different from zero, with zero being the value of the ideal case. All other parameters are for the reflected channel as described for the direct channel in Eq. (9).

Figure 4Normalized Fizeau reflection depending on commanded laser frequency obtained from an ISR measurement performed on 10 October 2018 (black dots) and the corresponding best fit of Eq. (10) (light-blue line).

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To investigate the ALADIN instrumental alignment and ongoing spectral drifts, a fit of Eqs. (9) and (11) to the ISR measurement data is performed by using a downhill simplex optimization method implemented in OriginLab. The sum of the Fourier series describing the Airy function is calculated for 51 terms. Considering a mean plate reflectivity of 0.65, the neglected terms only contribute to about 0.65⁵¹ = 3 × 10⁻¹⁰. Except for ℱ_FSR = 10 946 MHz and d_Fiz=0.5, all parameters are not constrained and thus a result of the fit routine. The assumption of a constant FSR is justified by the solid arrangement of the FPIs and the temperature stabilization of down to 10 mK, which results in a rather constant plate spacing. Even alignment changes that may alter the incidence angle on the FPI by for instance 5 mrad would change the FSR by only 0.1 MHz.

Based on the determined fit parameters, further quantities that characterize the FPI transmission curves can be derived. The FWHM of an ideal FPI is introduced by Eq. (6). After introducing a defect parameter that takes into account 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 ($\mathrm{\Delta }{f}_{{\text{FWHM}}_{\text{tot}}}$) is derived to be

$$\begin{array}{}\text{(12)}& \begin{array}{rl}\mathrm{\Delta }{f}_{{\text{FWHM}}_{\text{tot}}}=& \mathrm{0.53431}\mathrm{\Delta }{f}_{{\text{FWHM}}_{\text{ref}}}\\ & +\sqrt{\mathrm{0.21686}\mathrm{\Delta }{f_{{\text{FWHM}}_{\text{ref}}}}^{\mathrm{2}}+\mathrm{\Delta }{f_{{\text{FWHM}}_{\text{def}}}}^{\mathrm{2}}},\end{array}\end{array}$$

where $\mathrm{\Delta }{f}_{{\text{FWHM}}_{\text{ref}}}$ is given by Eq. (6), and $\mathrm{\Delta }{f}_{{\text{FWHM}}_{\text{def}}}=\mathrm{2}\sqrt{\mathrm{2}\ln \mathrm{2}}{\mathit{\sigma }}_{\mathrm{g}}$, which accounts for the broadening by defects. Thus, $\mathrm{\Delta }{f}_{{\text{FWHM}}_{\text{tot}}}$ provides a good possibility of monitoring the characteristic FPI transmissions without being influenced by the Fizeau interferometer.

4.2 Fizeau interferometer

In a Fizeau interferometer the plates are set with a wedge angle and spacing chosen to match the spectroscopic problem. The 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 are aligned parallel to the wedge vertex. A textbook analysis of the particular characteristics of the Fizeau interferometer is given by Born and Wolf (1980), drawing on the analyses of Brossel (1947), and has since been expanded upon 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 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 modeled 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

$$\begin{array}{}\text{(13)}& {\mathcal{T}}_{\text{Fiz}}\left(x\right)={\mathcal{I}}_{{\text{Peak}}_{\text{height}}}\left({\displaystyle \frac{\mathrm{\Delta }{f_{{\text{FWHM}}_{\text{Fiz}}}}^{\mathrm{2}}}{\mathrm{4}(x-x_{\mathrm{0}}{)}^{\mathrm{2}}+\mathrm{\Delta }{f_{{\text{FWHM}}_{\text{Fiz}}}}^{\mathrm{2}}}}\right),\end{array}$$

where ${\mathcal{I}}_{{\text{Peak}}_{\text{height}}}$ is the peak amplitude; $\mathrm{\Delta }{f}_{{\text{FWHM}}_{\text{Fiz}}}$ the FWHM; and x₀ the position of the fringe, which is usually called the 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 halfway 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, ${\mathcal{I}}_{{\text{Peak}}_{\text{height}}}$, $\mathrm{\Delta }{f}_{{\text{FWHM}}_{\text{Fiz}}}$, and the Mie response x₀ are available in the AUX-ISR product files.

Furthermore, the Fizeau transmission is calculated at each frequency step as the intensity sum of all 16 px 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 is applied for the Rayleigh signal is not considered for the Mie signal. The evaluation of the Fizeau transmission hence gives an approximation of potential changes in the beam intensity profile or beam diameter in one dimension.

Figure 5(a) FPI transmission curves of the direct channel (light-blue dots) and the reflected channel (yellow dots) measured on 10 October 2018, laser-energy-corrected by means of $E_{{\text{norm}}_{\text{new}}}\left(f\right)$ (see also Eq. 2), and the corresponding best fits according to Eqs. (9) and (11) in blue (direct channel) and orange (reflected channel), respectively. (b) Related relative residuals of the direct channel (blue) and reflected channel (orange). (c) Corresponding Mie response, i.e., the fringe center position on the ACCD pixel.

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Table 3Fit parameters according to Eqs. (9), (11), and (12) for energy-corrected ISR data by means of $E_{{\text{norm}}_{\text{new}}}\left(f\right)$. The given uncertainty in the fit values denotes the standard error derived by the fit routine.

The fit values in brackets denote the one retrieved from the data set corrected with E_norm(f) (see also Sect. 3.2). ^a Direct channel is at lower frequencies. The second crossing point is calculated by considering the measured FSR. ^b Values as given by Reitebuch et al. (2009). ^c Values taken from internal documents of the manufacturer that are not available to the public. NA: no specification available.

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4.3 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 Eqs. (9) and (11), the corresponding relative residuals and the derived Mie response are shown in Fig. 5. The corresponding fit results are summarized in Table 3. The given uncertainty in the fit values denotes the standard error derived by the fit routine, and the last column indicates the specification values for comparison.

In Fig. 5a, 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 $E_{{\text{norm}}_{\text{new}}}\left(f\right)$ (see also Eq. 2). The corresponding best fits according to Eqs. (9) and (11) are depicted by the blue and orange lines. Additionally, the frequency used for wind measurements (wind mode) is indicated by the dashed magenta line. The different peak transmission of both FPIs is a result of the sequential arrangement of the interferometers. This circumstance as well as other particular features such as the intensity dip (reflected channel, ≈ −1.2 GHz) and the modulation caused by the reflection on the Fizeau interferometer is adequately represented by the best-fit curves. The mean intensity per FSR is determined by the fit to be 3722 LSB (direct channel) and 3120 LSB (reflected channel), leading to an intensity transmission ratio of 0.84. The respective center frequencies are derived to be −1.239 GHz (direct channel) and 4.217 GHz (reflected channel), leading to a spectral spacing of 5.456 GHz when having the direct channel located at lower frequencies. The corresponding spectral spacing with the reflected channel being located at lower frequencies can be calculated by using the FSR according to 10.946 GHz − 5.456 GHz = 5.490 GHz. Thus, it is almost similar on both sides of the transmission peaks and differs by only 34 MHz. Furthermore it can be seen that the derived spectral spacing on both sides of the filter curves is well within the specification of (5477 ± 120) MHz.

Using the mean plate reflectivity (direct channel: 0.651; reflected channel: 0.652) and the defect parameter (direct channel: 147 MHz; reflected channel: 147 MHz) determined by the fit, $\mathrm{\Delta }{f}_{{\text{FWHM}}_{\text{tot}}}$ of the respective transmission curve can be calculated by means of Eq. (12) to be 1.589 GHz (direct channel) and 1.580 GHz (reflected channel). Thus, the FWHMs of the transmission curves are almost identical when neglecting the imprint of the reflection on the Fizeau interferometer and are smaller than the specified value of (1.667 ± 7) GHz. The numerically determined FWHM from the measured transmission curves however is 1.489 GHz (direct channel) and 1.711 GHz (reflected channel), showing that the Fizeau reflection can change the actual FWHM by several hundred megahertz. Additionally it can be seen that the spectral crossing point of the FPIs is at 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. 5b, the relative residual of the direct and the reflected channel as well as linear fits to the data is 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 per cent in amplitude and very constant 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 in 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, which 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 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 of measuring 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, for example, ≈ 1 ms⁻¹. This 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 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 mentioned in Sect. 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 were corrected with $E_{{\text{norm}}_{\text{new}}}\left(f\right)$ and for data that were 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 and 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.

The Mie response (see also Eq. 13) is shown in Fig. 5c. It can be seen that about five FSRs of 2.2 GHz are measured during an ISR (see also vertical dashed black lines that indicate one FSR). The actual range that can be used for wind measurements (useful spectral range, USR) is represented by the range where the Mie response is almost linear, i.e., about ± 0.6 GHz around each USR center frequency or, for example, the FPI filter crossing 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).

Figure 6Normalized FPI transmission curves acquired during ISR measurements performed from 10 October 2018 to 15 March 2021 (see also Table 2) for the FM-A period (October 2018 to June 2019, a) and FM-B period (July 2019 to March 2021, b).

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As these kinds of ISR measurements have been performed on a regular weekly basis, they offer the opportunity to analyze time series of the discussed fit parameters and with that to investigate spectral drifts and a change in the alignment conditions of the instrument as discussed in Sect. 5.

5.1 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 the mean transmitted intensity per FSR (ℐ_dir and ℐ_ref), which provides information about the signal levels on the ACCD detector for the internal reference path. It is worth mentioning that ℐ_dir and ℐ_ref do not consider the impact of the reflection on the Fizeau interferometer. Thus, the real mean intensity, calculated by integrating the actual transmission curves, would slightly differ from the derived fit values. However, an independent analysis (not shown) has verified that the obtained values are comparable, and the overall trend is similar for the different analyses. The mean intensities per FSR determined for the ISR measurements obtained from 10 October 2018 until 15 March 2021 (see also Table 2) are shown in Fig. 7a for the direct channel (blue circles) and the reflected channel (orange circles), respectively. The corresponding moving averages of five successive data points are indicated by the light-blue line and yellow line. Figure 7b shows the corresponding intensity ratio (${\mathcal{I}}_{\text{ref}}/{\mathcal{I}}_{\text{dir}}$, black circles) as well as the moving average of five successive data points (gray line). In Fig. 7c, the mean laser pulse energy during the respective ISR measurements retrieved from PD-74 is plotted. Special time periods as for instance an instrument shut off due to a GPS reboot anomaly (1), the switch from laser FM-A to FM-B (2), and a laser CPT optimization period (3) are indicated by gray bars.

The different periods with the respective lasers FM-A and FM-B can clearly be distinguished as the FM-A laser was operating with about 30 % lower laser pulse energy compared to FM-B (beginning of life), or more precisely, the ACCD detected 30 % less signal on the internal reference path. During the FM-A period, the mean intensity per FSR continuously decreased for both channels except for certain periods that were related to laser parameter optimizations (e.g., pre-amplifier and amplifier current changes in mid-November 2018). The decrease was almost linear in the beginning (October until November 2018 but also later on, from March until June 2019) and is determined to be (−4.7 ± 0.5) LSB d⁻¹ for the direct channel and (−4.4 ± 0.5) LSB d⁻¹ for the reflected channel in the time period from March to June 2019. Considering the initial intensity values of this time period of 3685 LSB (direct channel) and 3084 LSB (reflected channel), this corresponds to a decrease rate of 0.13 % d⁻¹ and 0.14 % d⁻¹, respectively. It is worth mentioning that the analysis of the PD-74 mean laser energy trend in that time period (Fig. 7c) indicates a larger decrease rate of about 0.18 %. This could for instance be explained by alignment changes that led to more photons transmitted through the internal optical path, hence partly compensating the actual laser energy decrease. The intensity ratio (Fig. 7b) changed from about 0.85 to 0.82 during the FM-A period. This change could be due to slightly different alignment changes in the direct-channel and reflected-channel optical paths that influence the transmitted photons differently (e.g., by clipping on other optical elements) and thus change the intensity transmission ratio or due to photons lost outside the ACCD differently for the two channels. It is interesting to realize that the trend of the intensity ratio almost continuously proceeds even when switching from FM-A to FM-B (June 2019), although the overall intensity levels remarkably increase. Furthermore from the bottom panel it can be seen that the overall mean laser pulse energy decreases from about 57.5 mJ in October 2018 to about 40 mJ in June 2019, which corresponds to a decrease of 30 % in the mentioned time period.

Comparing the signal levels at the beginning of the FM-A period (September 2018) and FM-B period (July 2019), an increase of about 37.5 % was achieved (i.e., from 4000 to 5500 LSB for the direct channel). As for the FM-A period, the mean intensity per FSR is continuously decreasing with FM-B. In September 2019, the decrease rate was (−3.7 ± 0.4) LSB d⁻¹ for the direct channel and (−2.6 ± 0.3) LSB d⁻¹ for the reflected channel and thus about 40 % smaller than for the FM-A period. Furthermore it can be seen that the laser CPT change that was performed in March 2020 led to a decrease in the mean intensity levels by about 15 % (i.e., from 4380 to 3740 LSB for the direct channel) but also to a remarkable reduction in the decrease rate, namely to (−1.9 ± 0.1) LSB d⁻¹ for the direct channel and (−1.5 ± 0.2) LSB d⁻¹ for the reflected channel. Interestingly, the mean laser pulse energy (Fig. 7c) 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 have become more variable due to appearing signal fluctuations in the internal reference 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⁻¹ for the direct channel and (−9.9 ± 0.3) LSB d⁻¹ for the reflected channel. After the laser parameter changes in December 2020, the intensity ratio also 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 the FPI center frequencies, as discussed in the next section.

Figure 8(a) Center frequencies for the direct channel (blue) and the reflected channel (orange) derived by fitting Eqs. (9) and (11) to the ISR data sets obtained from October 2018 to March 2021 (see also Table 2). Corresponding line fits to distinct time periods (period 1: 17 October 2018 to 9 January 2019; period 2: 15 February to 13 June 2019; period 3: 25 July 2019 to 2 March 2020; period 4: 30 March 2020 to 15 March 2021) are shown in light blue and yellow, respectively, and the derived slopes ($\mathrm{\Delta }{f}_{\mathrm{0}}/\mathrm{\Delta }t$) are given by the insets. (b) Corresponding spectral spacing Δf₀ between the direct channel and the reflected channel of the FPIs (black circles) and an exponential decay fit (gray) to show the settlement of the spacing drift. Special time periods are indicated by gray bars (1: instrument shut off due to a GPS reboot anomaly; 2: switch from FM-A to FM-B; 3: laser CPT optimization).

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5.2 FPI center frequencies and spectral spacing

The center frequencies derived by fitting Eqs. (9) and (11) to the ISR data sets obtained from 10 October 2018 until 15 March 2021 (see also Table 2) are shown in the top panel (a) of Fig. 8 for the direct channel (blue circles) and the reflected channel (orange circles), respectively, whereas the left y axis denotes the direct-channel frequencies and the right y axis the one of the reflected channel. As the derived center frequencies show a larger jump of 5.4 GHz between the FM-A period (September 2018 until June 2019) and the FM-B period (June 2019 until now), the y axes furthermore include a break of the same size. For a better comparability, the frequency range is chosen to be equal for both channels and both the upper and lower part of the panel, namely 150 MHz. Furthermore, line fits to four distinct time periods with almost-linear center frequency drift are shown by light-blue and yellow lines, respectively, whereas period 1 lasts from 17 October 2018 to 9 January 2019, period 2 from 15 February to 13 June 2019, period 3 from 25 July 2019 to 2 March 2020, and period 4 from 30 March 2020 to 15 March 2021. Corresponding slopes of the line fits ($\mathrm{\Delta }{f}_{\mathrm{0}}/\mathrm{\Delta }t$) are given by the insets. The overall frequency changes for the direct and the reflected channel within the said time periods are 22 and −62 MHz (period 1), 17 and −13 MHz (period 2), 49 and 34 MHz (period 3), and 47 and 53 MHz (period 4). These values are additionally summarized in Table 4. It is worth mentioning that the base laser frequency of 844.961832 THz is not measured but calculated from the laser vacuum wavelength of 354.8 nm that was determined during on-ground tests (Mondin and Bravetti, 2017). The y axis thus shows the commanded laser frequency value plus the laser base frequency. Hence, the 5.4 GHz frequency jump stems from a different absolute frequency of FM-B and could even be a few FPI FSRs different.

Figure 8b shows the spectral spacing between the two FPIs, which is defined as the spectral distance between the center frequencies of the direct channel and the reflected channel according to Δf₀ = $f_{{\mathrm{0}}_{\text{dir}}}-f_{{\mathrm{0}}_{\text{ref}}}$, where $f_{{\mathrm{0}}_{\text{dir}}}$ and $f_{{\mathrm{0}}_{\text{ref}}}$ are the center frequencies of the direct channel and the reflected channel, respectively, and $f_{{\mathrm{0}}_{\text{dir}}}<f_{{\mathrm{0}}_{\text{ref}}}$. The spectral spacing for $f_{{\mathrm{0}}_{\text{ref}}}<f_{{\mathrm{0}}_{\text{dir}}}$ can be calculated according to ℱ_FSR−Δf₀. The gray line indicates an exponential fit to the data set of FM-A (October 2018 to June 2019) to visualize the settlement of the spectral spacing evolution, and the magenta line depicts a temperature time series measured at the ALADIN detection electronic unit (DEU) as it is a good proxy for the ambient temperature within the ALADIN instrument. Special time periods as for instance an instrument shut off due to a GPS reboot anomaly (1), the switch from laser FM-A to FM-B (2), and a laser CPT optimization period (3) are indicated by gray bars.

It can be seen that the center frequencies of both channels are drifting with time by a few megahertz per week throughout the mission. This would not necessarily affect the Aeolus wind retrieval as long as the drift would occur for both channels in the same spectral direction and with the same rate. However, this is not inevitably true in the case of ALADIN. For the FM-A period for instance, it can be recognized that f₀ of the respective channels was drifting in different spectral directions and with a different rate. For period 1, the center frequency drift $\mathrm{\Delta }{f}_{\mathrm{0}}/\mathrm{\Delta }t$ was (0.26 ± 0.03) MHz d⁻¹ for the direct channel and (−0.73 ± 0.03) MHz d⁻¹ for the reflected channel. Thus, the center frequency drift of the reflected channel was about a factor of 2.8 larger and with different sign compared to the one of the direct channel. In the later FM-A phase (period 2) the drift rate settled and equalized to (0.14 ± 0.01) MHz d⁻¹ for the direct channel and (−0.15 ± 0.02) MHz d⁻¹ for the reflected channel, still having a different sign. For FM-B the situation is different. Here, the center frequencies of both channels drift 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⁻¹ (direct channel) and (0.16 ± 0.01) MHz d⁻¹ (reflected channel), whereas for period 4 it was (0.14 ± 0.01) MHz d⁻¹ (direct channel) and (0.16 ± 0.01) MHz d⁻¹ (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. 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 FPIs is the spectral spacing, as shown in Fig. 8b (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 is almost true for the FM-B period) would not affect the wind retrieval. At the beginning of the mission, the 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. 8b, gray line) applied to the FM-A data set that indicates an asymptotic convergence at 5320 MHz, which is about the spacing determined for FM-B in early 2021. This finding would lead us to expect that the drifting optical element is not located in the laser transmitters but between the laser transmitter and optical receiver bench, or it would allow 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 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) are obvious, whereas the temperature changes are caused by entering or leaving a solar eclipse phase, where parts of the satellite orbit are in darkness, leading to a temperature decrease in 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 temperature of the FPIs only changes by 10 mK during these eclipse periods (not shown).

Figure 9Total FWHM of the direct channel (blue circles) and the reflected channel (orange circles), derived by fitting Eqs. (9) and (11) to the ISR data sets obtained from 10 October 2018 until 15 March 2021 (see also Table 2) and calculated according to Eq. (12). Special time periods are indicated by gray bars (1: instrument shut off due to a GPS reboot anomaly; 2: switch from FM-A to FM-B; 3: laser CPT optimization).

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5.3 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 Eqs. (9) and (11) to the measured FPI transmission curves. The FWHMs 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 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 FWHM increased rather constantly by about 60 MHz, whereas it only slightly decreased for the reflected channel, by about 20 MHz. When switching to FM-B, the obtained direct-channel FWHM shows a decrease of about 70 MHz, indicating different alignment conditions for both lasers. Interestingly, the jump is smaller for the reflected channel, namely about 20 MHz. At the beginning of the FM-B phase, the direct-channel FWHM shows an increase from about 1580 MHz to 1650 MHz, where it settles around December 2019. This is the same value that was reached at the end of the FM-A period. On the other hand, the reflected channel shows again a FWHM decrease at the beginning of the FM-B period and seems to settle in May 2020 at a value of about 1570 MHz. Since August 2020 the determined fit results from week to week have become more variable due to appearing intensity variations in the degrading internal reference signal transmission (see also Sect. 6).

Figure 10Fizeau center frequency derived from the direct channel (blue circles) and the reflected channel (orange circles) by fitting Eqs. (9) and (11) to the ISR data sets obtained from October 2018 to March 2021 (see also Table 2). The corresponding moving average of five successive data points is indicated by the light-blue line and yellow line, respectively. Special time periods are indicated by gray bars (1: instrument shut off due to a GPS reboot anomaly; 2: switch from FM-A to FM-B; 3: laser CPT optimization).

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5.4 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 $f_{{\mathrm{0}}_{\text{Fiz}}}$ of the Fizeau reflection is a free fit parameter for the model functions describing the FPI transmission curves. The values determined by fitting Eqs. (9) and (11) to the ISR data sets obtained from October 2018 to March 2021 (see also Table 2) are shown in Fig. 10 for the direct channel (blue circles) and the reflected channel (orange circles), respectively. The corresponding moving averages of five successive data points are indicated by the light-blue line and yellow line. Special time periods as for instance an instrument shut off due to a GPS reboot anomaly (1), the switch from laser FM-A to FM-B (2), and a laser CPT optimization period (3) are indicated by gray bars.

Figure 11Integrated Fizeau intensity versus Mie response obtained during ISR measurements (see also Table 2) normalized to unit area for the FM-A (a) and the FM-B (b).

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It can be seen that the Fizeau center frequency is continuously drifting during the FM-A period. In particular, the drift is about 100 MHz within 9 months. Additionally, it can be recognized that the derived trend of the Fizeau center frequency is rather similar for the direct and the reflected channel but has an offset of about 50 MHz, which could be due to the fact that the reflection on the Fizeau is not exactly periodical, and thus, the imprint on 𝒯_dir and 𝒯_ref is not exactly the same. Nevertheless, it can be concluded that the spectral imprint of the Fizeau reflection on the FPIs is drifting over time during the FM-A period.

During the FM-B period, the situation is different. After switch-on of FM-B, a Fizeau center frequency drift of about 80 MHz was observed until September 2019, followed by an obvious settlement. In December 2019 a drift of about 50 MHz was determined from the direct-channel analysis, whereas the reflected channel showed a drift by a similar amount but in a different spectral direction. This drift is related to a laser CPT jump that occurred on 9 December 2019. The CPT drift that happened afterwards is responsible for the Fizeau center frequency drift. With the settlement of the laser CPT, the derived Fizeau center frequencies also settled until about August 2020. Since then, the obtained fit parameters are in general more variable due to a larger variability in the spectrometer signals, which could be explained by beam clipping happening due to the ongoing alignment drift (see also Figs. 7 and 8). Besides the analyses of the Fizeau reflection impact on the Rayleigh channel signals, Mie channel signals are also 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.

5.5 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 intensity profile or diameter of the beam that is illuminating the Fizeau interferometer (in one dimension). It is worth mentioning that any changes in the measured Fizeau intensity are mainly caused by a variation in the interferometer illumination of the internal path rather than due to a change in the Fizeau transmission itself as the overall illumination is mainly determined by the laser beam profile due to the near-field image of the beam that is used within the fringe imaging technique. The integrated Fizeau transmissions versus Mie response (Eq. 13) obtained during ISR measurements performed from 10 October 2018 until 15 March 2021 (see also Table 2) normalized to unit area (to emphasize the beam shape change) are shown in Fig. 11 for the FM-A period (a) and the FM-B period (b). Brownish colors correspond to the early time of the respective laser period and blueish colors to the later times (see also the label of each panel).

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 decreases with time, which could be explained by a shrinking beam 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.