Experiment with simultaneous measurements at two optical wavelengths in the FG5X absolute gravimeter

The FG5X gravimeters are the most accurate commercially available absolute gravimeters at present. They traditionally use one optical wavelength in their interferometer to measure the gravity acceleration of the freely falling test mass in a vacuum. In this paper, for the first time, it is demonstrated the possibility to track the test mass simultaneously with two optical wavelengths of 633 nm and 771 nm and to evaluate the gravity acceleration from both these measurements. We show the technical solution, mathematical methods and error sources that have to be taken into account for the realization of simultaneous interferometric measurements. The achieved results show agreement of the gravity accelerations at the level of 2–4 microgals and increased low-frequency noise in residuals at 771 nm due to optical optimisation of a gravimeter to the wavelength of 633 nm. We evaluated the sensitivity of gravity measurements to the used wavelength as a new contribution in the uncertainty budget that for the FG5X gravimeter reached 0.25 μGal and 2 μGal at wavelengths of 633 nm and 771 nm, respectively. Further, we discuss that the optimisation of a gravimeter to a certain wavelength is related to the applied antireflective coating on the optical elements of the gravimeter among them the glass retroreflector plays the key role since its movements being dominant.


Introduction
The accurate absolute gravity measurements are mainly used in geosciences (e.g.geodesy, geodynamics, volcanology, Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.hydrology), contributing to monitoring of the Earth system [1].In metrology, the highest accuracy requirements on gravity measurements are related to the realization of the kilogram by the Kibble balance, where the contribution from absolute gravity measurements should not exceed the standard uncertainty of 5 µGal [2] (1 µGal = 10 nm s −2 ).The third International Comparison of Absolute Gravimeters held at the BIPM in 1989 confirmed that the reference wavelength is one of the most important contributions in the uncertainty budget of the gravity measurements [3].At that time, the problem was solved by using an iodine stabilized He-Ne laser.The FG5 [4] and FG5X [5] types of gravimeters (FG5/X), the most accurate commercially available absolute gravimeters at present, follow this concept and they use Winters Electro Optics (WEO) He-Ne laser stabilized to an iodine transition.The FG5/X gravimeters, reaching an uncertainty of about 2-3 µGal, dominate in the comparisons [6] organized under the International Committee for Weights and Measures (CIPM) mutual recognition arrangement (MRA).Thus, it is important to analyse potential systematic errors of these types of gravimeters and carefully elaborate the associated uncertainty budget.Several error sources of FG5/X gravimeters have been detected in the past as the self-attraction [7] and recently the diffraction [8], distortion [9] and verticality effects [10].
The FG5/X gravimeters are using the frequency stabilized He-Ne laser at 633 nm as the traceability source for the length measurements.An experiment with the FG5 gravimeter was demonstrated in [11], where the authors used the solid-state laser at wavelength of 532 nm mounted on the gravimeter body and compared the results with the standard configuration at 633 nm, reaching an agreement within 2 µGal for sequential measurements and not reporting problems related to the optimisation of a gravimeter for different wavelengths.However, they also do not report how the diffraction effect was treated for both wavelengths.
The aim of our experiment, described in this paper, was to investigate whether the FG5X gravimeter could be operated by a frequency stabilized laser working at a wavelength of 771 nm.Further, we aimed to investigate if our measurement system, working with the FFT swept filtering method [12], is capable to analyse measurements at two wavelengths simultaneously.The advantage of the FFT swept filtering method, as a digital method applied to the electrical signal of the optical interferometer, is that it minimizes errors from the electrical and optical analogue filters.The advantage of simultaneous measurements is that the gravity variations are suppressed in differences.The fibre optics of the FG5X gravimeters allowed us to use different collimators during these experiments and to achieve interesting results related to the diffraction effect, which has a significant contribution in the uncertainty budget of the FG5X gravimeters.The increased low frequency noise in residuals at 771 nm forced us to deal with the unpublished gravimeter optimisation for the certain wavelength and to evaluate its sensitivity for particular wavelengths.
In section 2, we describe the measurement method that allowed to carry out the gravity measurements simultaneously by two lasers with different optical wavelengths and by using two collimators optimised for corresponding wavelengths.The next section deals with corrections that have to be taken into account during such simultaneous measurements.In section 4, the measurement noise is compared for the 771 nm and 633 nm interferometers, facing the problem of optimisation of the FG5X gravimeter to the certain wavelength.In section 5, the measurement results for the gravity acceleration are presented and discussed.Finally in section 6, we are estimating and discussing the effect of optimisation in relation to the antireflective (AR) coating.

Measurement setup and evaluation principle
The measurements described below were carried out with the FG5X-251 gravimeter that uses the WEO model 100 iodine-stabilized He-Ne laser working at the wavelength of λ = 633 nm.Standardly, a single-mode polarizationmaintaining optical fibre is used to deliver the laser radiation into the collimator on the input of the interferometer.
In addition to the 633 nm laser, we used the Stabiλaser 1542ε (epsilon edition) that is the second edition of the acetylene-stabilized laser described in [13].We selected its second-harmonic output at 771 nm for the gravity measurements to obtain fringes by the silicon-based detector common to both wavelengths.The 771 nm wavelength is not frequency modulated.Nevertheless, we verified that the frequency modulation in the fitting procedure does not affect the evaluation of gravity measurements for unmodulated lasers.A fibre coupler was used to deliver two lasers into the FG5X-251 gravimeter.The measurement setup of the gravimeter working simultaneously with two laser wavelengths is shown in figure 1.
The gravity measurements were carried out with both lasers present in the interferometer at the same time (i.e.simultaneously) as well as measurements with both lasers separately.The physical separation of radiation was technically done by a fibre-decoupling of individual lasers.The gravity measurements with two simultaneously-coupled lasers cannot be analysed by the original fringe-detection system of the gravimeter, because it uses only the zero-crossing-based fringe counting, which is affected by the second wavelength.However, we can use the second detection system of the FG5X-251 gravimeter based on the FFT swept filtering method of the signal acquired by the HS5 oscilloscope [12].This system, mounted with an additional photodetector in the complementary output of the interferometer, could work in parallel with the original detection system.The FFT swept filtering method allows a software-based separation of the interferometric signals from two lasers with sufficiently different wavelengths [9].The principle of such a wavelength separation is shown in figure 2. The interferometric signal during the free fall is divided into the FFT blocks (see also [12]).Each block has its band-pass filter with the centre frequency that linearly increases with the time of free fall and frequency of the laser radiation.The filtered fringe signal is then reconstructed by the inverse FFT transformations separately for the 771 nm and 633 nm wavelengths.Finally, the gravity acceleration is evaluated from the 771 nm and 633 nm fringe signals by the standard method using two different reference vacuum wavelengths.
Note that the wavelength separation of signals is generally not possible at the top of the free fall, where the fringe frequency difference is zero, and thus the bandwidth of the filter must be also zero for the separation.However, the bandwidth cannot be close to zero due to the Fourier transform limit.Nevertheless, the fringe signal is not usually evaluated too close to the top of the free fall.Therefore, we use a delay of about 31 ms from the top of the free fall as the beginning of the evaluation interval for the FG5X gravimeter.One reason of this practice is that the electronic dispersion of the fringe signal is large at low frequencies [14], which affects the evaluation of the free fall acceleration.

Corrections of interferometric measurements
The laser radiation is delivered to the FG5X gravimeter via the optical fibre with the FC/APC connectors, which are anglepolished to prevent interferences due to the back reflections.The beam output from the fibre is then collimated by a collimator.As we showed in [8], the original collimator used in the FG5X gravimeters suffers from the lower beam quality.Thus, we did not used it as well as other reflective collimators, which are ideal for collimating multiple wavelengths in one device, but they have lower beam quality and the diffraction correction is more uncertain.Instead, we used two triplet collimators: (1) the triplet collimator optimised for the wavelength of 633 nm (Thorlabs TC25APC-633) that has been a reference collimator for the FG5X-251 gravimeter since 2017, (2) the triplet collimator (Thorlabs TC25APC-780) optimised for the wavelength of 780 nm that is close to the wavelength of the Stabiλaser (771 nm).
As shown in [8], the estimation of the diffraction correction for the triplet collimator, with a good beam quality factor, does not need a special approach, because the estimates agree within the measurement uncertainty.Therefore, the use of triplet collimators in the described experiment allowed us to evaluate the diffraction correction by an approach, which takes into account the evolution of the beam width with the propagation distance.The beam widths were measured by using a camera placed at several distances from the beam output of the gravimeter and the uncertainty of the diffraction correction has been estimated as 0.5 µGal.The main contribution to this uncertainty is associated with the beam quality factor that is similar for the 633 nm and 780 nm triplet collimators.The contribution from different sensitivities on two close wavelengths is playing negligible role in our case.The diffraction correction evaluated for both the triplet collimators and both laser wavelengths can be found in table 1 together with the differences between two wavelengths for each collimator.These differences are representing the expected differences, which should, in an ideal case, correspond with the differences between the gravity accelerations obtained from the simultaneous measurement of the gravimeter with two wavelengths.
The second expected difference in the simultaneous gravity measurements with two wavelengths is caused by the collinearity error.The parallelism of the interferometric beam with the local vertical is usually established for the FG5X gravimeters by the alcohol pool method, which however could be influenced by non-negligible errors as it was shown in [10].Therefore, we used a more accurate and bias-free method described in [10].In this method, the test beam is monitored laterally by a position sensitive detector.The alignment of the verticality is realized by minimizing of the evaluated accelerations of the beam spot during the free fall, corresponding to the condition that the test mass does not accelerate laterally with respect to the laser beam direction.If such an alignment is reached, the laser beam is vertical because there are negligible horizontal accelerations of the test mass during the free fall.
The measurements of the test mass accelerations evaluated laterally with respect to the laser beam directions after the verticality alignment are shown in figure 3 for the setup with the 780 nm collimator.As it can be seen, there is a deviation between both beams (at the 633 nm and 771 nm wavelengths) due to their natural non-collinearity as they exit from the same spectrally-imperfect collimator.We intentionally adjusted lateral accelerations of both beam spots roughly  to opposite sides from the perfect verticality (zero accelerations) to minimize the difference between the verticality errors for both beams.The lateral acceleration of the test mass 0.1 mm s −2 corresponds to the verticality error of the measured acceleration of about −0.05 µGal as the cosine error of the angle between the laser beam direction and the vertical due to the gravity (see [10] for details).Thus, as it can be seen from figure 3, the verticality errors can be still neglected in our measurements with the optimal verticality alignment.

Measurement noise
Two wavelengths simultaneously present in the interferometer of the gravimeter allow a direct comparison of the measurements between two laser sources during the free fall.The residuals of the free fall motion are shown in figure 4. They clearly show that there is a drop-to-drop irreproducibility of the free fall measurements with the 771 nm laser, even though the 780 nm collimator was used for these measurements.This is probably due to the fact that the interferometer of the FG5X gravimeter is optimised for its usual wavelength of 633 nm.The residuals of the proper interferometric signal can be affected by parasitic interferences caused by reflections, which are reduced by the AR coatings.The parasitic interferences are proportional to the residual reflectance and they modify the correct phase of proper interferometric signal.Thus, the observed low-frequency interferometric noise degrades the gravity measurements to some extent for the 771 nm laser.The averaged residuals from 300 drops in figure 4 show that the use of a non-optimised wavelength in a FG5X gravimeter also causes a setting-dependent low-frequency waves in the residuals of the free fall with the amplitude of about 100 pm.
According to [15], such a wave in the averaged residuals indicates that also the averaged gravity accelerations might be affected.Therefore, an additional contribution of 2 µGal has been added to the uncertainty budget.This contribution includes the sensitivity of g estimate on the selected evaluation interval.The simultaneous measurements in figure 4 indicate that the low-frequency waves have optical origin.We also carried out a short measurement at the fundamental wavelength of 1542 nm from the Stabiλaser 1542ε.The measured residuals show even larger low-frequency waves (see figure 5).It indicates that the interferometer is even more optically non-optimised to the 1542 nm wavelength than to the 771 nm wavelength.The gravimeter sensitivity to this non-optimisation generally depends on the reflectance of the AR coatings on optical elements that the laser goes through.The spectral reflectance curve of the used AR coatings cannot be verified by the gravimeter users, but it is generally a nonlinear function as it will be shown in section 6.
The discussed low-frequency waves in simultaneous measurements are clearer from the spectra of the residuals (see figure 6), where similar acoustic peaks measured by both interferometers can be identified.The distinction between measurement noise sources is only made possible by the simultaneous measurements.The spectra of the residuals also show that the high-frequency noise is larger for the 633 nm laser that has a worse stability of the lasing frequency than the 771 nm laser.The frequency stability of both types of lasers is shown in figure 7. The Stabiλaser is more frequency stable than the WEO laser also for time averages comparable to the free fall duration (0.25 s).The short-term frequency stability of both lasers is capable of reaching accuracy and precision of g at the level better than 1 µGal, however, it is just a necessary but not a sufficient condition for the stability of the gravity measurements at this level.
It is important to note that in reality the gravity measurements with the FG5/X gravimeters are relatively more sensitive to the stability of the length reference rather than of the time reference due to the significant dead path of the FG5/X interferometer.It is because the length difference between two interferometric arms is of about two meters and these are relevant to the changes in the measured position of the test mass in terms of the optical wavelength.However, the timing of the free fall corresponds directly to the displacement that is about one order of magnitude lower than the difference between the interferometric arms.However, the interferometric measurements also depend on the purity of the interferometric signal.Possible parasitic interferences could affect the interferometric measurements more than the stability of optical wavelength itself.

Gravity measurements
The stability of the gravity acceleration for simultaneous measurements with two wavelengths are shown in figure 8.A larger noise can be seen for the wavelength 771 nm as a consequence of the larger low-frequency noise in the interferometric residuals shown in the previous section.In addition, the stability of gravity measurements at 1542 nm is also included in figure 8, showing almost one order larger noise.Note that this increase cannot be attributed to the gravity acceleration itself, although measurements at 1542 nm have not been done simultaneously with the 633 nm and 771 nm wavelengths, because the noise of gravity was the same during all these measurements according to continuous observations with a superconducting gravimeter at the station.The measurements at 1542 nm will be further discussed in section 6, including the explanation why the corresponding absolute gravity value cannot be used for a comparison, and therefore will not be discussed in this section.
The simultaneous gravity measurements with two wavelengths (771 nm and 633 nm) were carried out for two setups by using two collimators (780 nm and 633 nm) mounted in the interferometric input of the FG5X gravimeter.The corresponding gravity differences are compared in table 2 with and without the corrections for the diffraction effect according to table 1.It can be seen that in case of the applied diffraction corrections, the gravity differences are significantly reduced.However, this result has to be introduced with associated uncertainties, where three contributions have to be taken into account: (1) the experimental standard deviation of the mean  differences from 300 free falls that is below 1 µGal, (2) the uncertainty of 0.5 µGal for the applied diffraction corrections [8] (3) the uncertainty of µGal from potential due the low-frequency wave in the residuals (figure 4(a)).Then, the total expanded uncertainty of the differences is of about 4.5 µGal, covering the measured gravity differences obtained from the simultaneous measurements with the 771 nm and 633 nm laser wavelengths.
In addition to the simultaneous measurements, the gravity measurements in the classical configuration with the 633 nm laser only and with the 633 nm collimator were carried out just few days before the measurements described above and the non-instrumental gravity variations has been monitored by a superconducting gravimeter and taken into account.The absolute gravity value corrected for all instrumental effects including the diffraction reached 980 930 188.6 µGal and 980 930 189.1 µGal for the measurement system working with the FFT swept filtering method and the original system, respectively.The difference of −0.5 µGal is at the level of measurement repeatabilities of both systems.
It is worth to compare the absolute gravity value reached in the classical single-wavelength configuration with those reached during the simultaneous measurements that, however, could be evaluated only by the system with FFT swept filtering.The gravity value for the 633 nm laser and with the same 633 nm collimator reached 980 930 188.2 µGal for the  The Allan standard deviation of the gravity acceleration (corrected for geophysical effects) for simultaneous measurements (300 free falls with the time interval of 10 s) with two wavelengths (633 nm and 771 nm) supplemented with non-simultaneous measurements at 1542 nm (30 free falls with the time interval of 10 s).simultaneous measurement.It differs only by −0.4 µGal with respect to the result achieved in the classical configuration.It shows that a presence of an additional wavelength of 771 nm did not affect the measurement and its evaluation by the FFT swept filtering method.
Further, it is worth to specify that the simultaneous gravity measurements for the 771 nm laser with the 780 nm collimator give the diffraction-corrected result of 980 930 187.8 µGal.It differs from the absolute gravity value obtained for the 633 nm laser with the 633 nm collimator by −0.4 µGal that is also below the repeatability of measurements.

Effect of AR coatings
The measurements in [17] showed that the main parts of the FG5X gravimeter mutually do not change their position at a micrometre level during the free fall of the test mass.The differences between the gravity accelerations evaluated from the simultaneous measurements of the FG5X gravimeter with 771 nm and 633 nm laser wavelength for setups with the 780 nm and 633 nm collimators, respectively.The differences are introduced without and with the diffraction corrections applied according to table 1.The experimental standard deviation of the mean differences and the total expanded uncertainty of the corrected differences are shown in the third and fourth columns, respectively.This is self-evident also from the realized measurements with sub-nanometre precision, thus the gravimeter could be considered as sufficiently rigid.However, there is naturally an exception-the inertial reference and freely falling test mass with the glass retroreflector which translates and rotates much more than the inertial reference during the free fall.Thus, the most likely source of the signal from optical non-optimisation are the parasitic reflections from the glass retroreflector in the test mass depending on the effectiveness of the AR coating for a certain wavelength.
The interferometer in the FG5X gravimeter is designed to minimize a presence of parasitic beams.The vacuum window is tipped by 0.5 • to avoid parasitic reflections [4].Parasitic beams are then deflected, and thus further attenuated by limiting holes located along the beam path.However, the glass retroreflector can cause parasitic internal reflections.Such a parasitic test beam then passes a round trip inside the glass retroreflector, keeping the parallelism with the direction of the original ordinary test beam.Thus, it will also interfere with the reference beam, but its delay will depend on the rotation of the test mass during the free fall, according to [4] reaching of about 10 mrad s −1 and varying drop-to-drop.Therefore, such parasitic reflections could produce the low-frequency waves in residuals that also vary drop-to-drop to some extent due to the actual rotation of the test mass.However according to [18], 'the angular rotation of the proof mass can increase, as the contacts on the proof mass are worn or if the bearings on the release mechanism become loose with time.'Thus, the stochastic rotation speeds could also evolve with time.As a consequence, the sensitivity of the non-optimisation effect on gravity measurements cannot be considered as stable over time and, of course, it will be different for a particular gravimeter.On the other hand, this effect on measurements at different wavelengths can be used for detecting changes in the retroreflector rotation rates that cannot be measured in the FG5X gravimeter by the rotation monitor device that was available only for the FG5 gravimeters.
The finding that the measurements of the FG5X gravimeter could be influenced by optical reflections dependent on the used laser wavelength and spectral reflectance of AR coatings should be demonstrated by the gravimeter sensitivity to the optical optimisation.However, the previous section showed that for such a purpose, a single-frequency laser with the frequency stability better than 10 −10 in relative have to be used.It is not easy to find such lasers close to the 633 nm He-Ne laser.Nevertheless, as already discussed in sections 4 and 5, we had opportunity to carry out a short non-simultaneous measurement with the fundamental wavelength of 1542 nm from the Stabiλaser 1542ε.In this case, we had to use a near-infrared single-mode polarization-maintaining optical fibre instead of the fibre coupler of two visible wavelengths.Further, also the silicone APD detector needed to be replaced by the InGaAs PIN detector, and thus the simultaneous measurements with visible wavelengths were not possible.The diffraction correction for 1542 nm has not been evaluated since our camera could not be used at this wavelength range.Therefore, we were not able to evaluate the gravity acceleration with sufficient accuracy and the mean gravity value is significantly biased, and thus it is not presented in section 5. Nevertheless, the stability of gravity measurements at 1542 nm could provide useful insight on the effect of the optimisation.The measurement stability expressed by the Allan standard deviation is plotted in figure 8 along with the stability of gravity measurements for visible wavelengths.It can be seen that the stability at 1542 nm is about six times worse than for the second harmonic wavelength of 771 nm that is about two times worse than the overall stability of gravity measurements for the 633 nm wavelength.It must be noted that the frequency and amplitude stability of radiation of the fundamental wavelength of 1542 nm should be better or at least the same as for its second harmonic wavelength 771 nm.Thus, the worse interferometric stability at 1542 nm also indicates that the gravimeter optics are optimised for a certain wavelength.The AR coating of the glass retroreflector in the FG5X gravimeters is designed as the V-coating optimised for perpendicular reflections at 632 nm [19].The overall stability of gravity measurements is compared with a typical V-coating reflectivity curve in figure 9 showing a large and complex spectral dependence of the effect.
The uncertainty contribution of about 13 µGal was estimated from a potential error due to the low-frequency wave in the residuals at 1542 nm (figure 5) using [15] as in the case of 771 nm described above.Based on ratios between the reflectances at different wavelengths, we suppose that the gravimeter sensitivity on optical optimisation at the 633 nm will contribute to the uncertainty of gravity measurements at the level of about 0.25 µGal for a typical V-coating for 633 nm with the reflectance below 0.25% at 633 nm.Noise of the gravity measurements (expressed by the Allan standard deviation of individual drops and shown by dots) at a particular optical wavelength that have been used in the interferometer of the FG5X gravimeter A typical spectral curve of the reflectivity of anti-reflective V-coating at 633 nm according to Thorlabs [20] (it means not exactly those used for the FG5X gravimeter) is shown for a comparison on the secondary axis on the right.

Discussion and conclusions
Experimental measurements with two optical wavelengths in the absolute gravimeter have for the first time demonstrated a possibility for simultaneous measurement of the gravity acceleration by two wavelengths, showing also the capability of the FFT swept method for analyses of the fringe signal as well as to identify possible error sources of interferometric measurements at a particular wavelength.We have also demonstrated that the original system of the FG5X can use directly (with a minimal hardware modification) a different wavelength than the traditional 633 nm from the frequency stabilized He-Ne laser.Nevertheless, the fact that the optical properties of the FG5X interferometer are optimised for the 633 nm laser, not presented or communicated up to now, could be seen in the important low-frequency noise in residuals.During the experiment with the FG5X-251 gravimeter, we evaluated that it contributes to the uncertainty budget by about 0.25 µGal, 2 µGal and 13 µGal at the laser wavelength of 633 nm, 771 nm and 1542 nm, respectively.The AR coatings are the cause of this nonoptimised performance dependent on the reflectance at different wavelengths.Therefore, at present the gravity measurements with the FG5X gravimeter by using the 771 nm laser have a higher uncertainty and the gravimeter needs to be adapted (by the manufacturer) to benefit from the better performance of a laser such as the Stabiλaser 1542ε operating at 771 nm.One possibility to reduce the effect of optical reflections is to replace the glass retroreflectors by the mirror retroreflectors, which should have naturally zero effect for an arbitrary wavelength.Such a modification has recently been realized [19] in a few of the FG5X gravimeters and it would allow to verify our assumption that the main source of reflections is the freely falling retroreflector.On the other hand, in the unmodified gravimeter, the simultaneous measurements could be utilized for testing the rotation rates of the retroreflector, which are evolving in time with number of realized free falls and can have a non-negligible effect on the measurement of gravity acceleration.Nevertheless, it is important to note that our experiment showed that the gravity measurements with FG5X gravimeters are limited neither by the frequency stability of the 633 nm laser nor by the contribution of optical non-optimisation at 633 nm.
On average, the estimated gravity values achieved with the 771 nm laser and 633 nm laser agree within the measurement uncertainty for the FFT swept method.However, in case of the original detection system of the FG5X gravimeter, its dependence on the fringe amplitude [12] has to be taken into account because the power of lasers might be significantly higher than it is usual.Thus, lower power signals should be used for the original detection system to suppress this effect on the gravity values.
The advantages of the Stabiλaser with the second harmonic wavelength of 771 nm over the WEO laser with the standard wavelength of 633 nm could be summarized as: • the Stabiλaser internally uses optical fibres, and thus the output is directly coupled into the optical fibre of the FG5X gravimeter, whereas the WEO laser has the free space beam output that must be aligned into the optical fibre, which is less user-friendly, • the Stabiλaser has a lower frequency noise and it is a frequency-unmodulated laser (as, for example, the WEO model 200 laser at 633 nm), • the Stabiλaser has a higher optical power, and thus a lower detection noise is reached, • the WEO laser has the free space beam in the laser cavity, where optical surfaces occasionally need to be cleaned, • the Stabiλaser (Koheras BASIK laser module inside) could have a longer lifespan.
On the other hand, the disadvantage of the Stabiλaser is that the laser beam is not visible to the eye, but the gravimeter could be aligned with the help of a detector.The main disadvantage is, however, the above-mentioned issue with the optical optimisation of the FG5X gravimeter.
The described experiment carried out with the FG5X gravimeter for the first time elaborated simultaneous measurements with two wavelengths in a gravimeter based on laser interferometry.In addition to the mentioned limitations caused by the optimisation of a gravimeter for a specific wavelength, we showed approaches for the treatment of generally noncollinear and non-collimated beams that allowed to reach agreements between results of measurements.

Figure 1 .
Figure 1.The scheme describes the measurement setup of the FG5X gravimeter with two lasers fibre-coupled into one triplet collimator realising the input beam to the interferometer of the gravimeter.

Figure 2 .
Figure 2. The fringe signal from the HS5 detection system of the FG5X-251.(a) Example of one FFT block used by the FFT swept method.The FFT of the signal (blue line) is software-filtered to obtain separate signals for the 771 nm (dark red line) and 633 nm (red line) wavelengths.(b) The centre frequency of the band-pass filter sweeps linearly during the time of free fall.The bands of the FFT filter overlap at the beginning of the free fall.

Figure 3 .
Figure 3.The example of measurements of the lateral acceleration of the test mass with respect to the laser beams of the 771 nm (circles) and 633 nm (squares) interferometers for the same alignment of the gravimeter equipped with the 780 nm collimator.Each point represents results for one free fall realized by the FG5X-251 gravimeter.Small lateral accelerations (0.1 mm s −2 corresponds to the verticality error of about −0.05 µGal) indicates that the verticality error can be estimated with sufficient precision from less than 10 free falls.

Figure 4 .
Figure 4. Residuals of three consecutive free falls with the FG5X-251 gravimeter evaluated simultaneously by the 771 nm (a) and 633 nm (b) interferometers, respectively.Each free fall is indicated by the same colour.The average of 300 free falls is plotted in black.

P Křen et alFigure 5 .
Figure 5. Residuals of three consecutive free falls realized by the FG5X-251 gravimeter with the laser at 1542 nm (black lines) and compared with measurements at 771 nm (dark red lines) from figure 4(a).

Figure 6 .
Figure 6.The average of the FFT of the residuals obtained from 300 free falls simultaneously measured by two interferometers.

Figure 7 .
Figure 7.The Allan standard deviation of the optical frequency of the Stabiλaser 1542ε (s/n 2846) at 1542 nm/771 nm (error bars) and the WEO laser at 633 nm according to[16] for another device of the same type (filled squares) and the manufacturer specifications (hollow squares).

Figure 8 .
Figure8.The Allan standard deviation of the gravity acceleration (corrected for geophysical effects) for simultaneous measurements (300 free falls with the time interval of 10 s) with two wavelengths (633 nm and 771 nm) supplemented with non-simultaneous measurements at 1542 nm (30 free falls with the time interval of 10 s).

Figure 9 .
Figure 9. Noise of the gravity measurements (expressed by the Allan standard deviation of individual drops and shown by dots) at a particular optical wavelength that have been used in the interferometer of the FG5X gravimeter A typical spectral curve of the reflectivity of anti-reflective V-coating at 633 nm according to Thorlabs[20] (it means not exactly those used for the FG5X gravimeter) is shown for a comparison on the secondary axis on the right.

Table 1 .
The diffraction correction of the laser beams for different triplet collimators.