Comparison of fiber interferometric sensor with a commercial interferometer for a Kibble balance velocity calibration

This article presents a fiber interferometric sensor (FIS) for measuring the velocity amplitude of an oscillatory vibrating object, with a focus on velocity mode measurement in applications using the Kibble balance principle. The sensor uses the range-resolved interferometry method to measure the displacement of the moving object and employs a multi-harmonic sine-fit algorithm to estimate the displacement amplitude and frequency, thereby determining the velocity amplitude. This article provides a comprehensive explanation of the experimental setup and the measurement techniques employed, as well as a detailed analysis of the uncertainty budget, with the performance validation of the FIS benchmarked against a commercial interferometer within a Kibble balance setup. The velocity amplitude of a coil of the Kibble balance, oscillating with an approx. amplitude of 20 μm and a frequency of 0.25 Hz, was measured using the sensor and found to be 31.282 31 μm s−1 with a relative deviation of −1.9 ppm compared to a commercial interferometer. The high performance of the FIS, especially with regard to non-linearity errors, and the small size of the measuring head enable universality of integration into a wide variety of measurement systems, also including the use as general-purpose vibration and displacement sensor.


Introduction
Interferometric measurement techniques are widely used for dimensional measurement in high precision force measuring instruments, due to their ability to make non-contact measurements with high accuracy and, through knowledge of the * Author to whom any correspondence should be addressed.
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. laser wavelength, traceability to the meter definition [1,2]. For measuring dynamic displacement and acceleration signals for force transducer calibration, a heterodyne interferometer is commonly used [3,4]. On the other hand, homodyne interferometer techniques are typically employed for measuring static or low frequency signals in precise force calibration or measurement instruments, such as calibrating the forcedisplacement curves (spring constant) of AFM cantilevers [5].
Interferometers play a crucial role in applications where the traceability of a measuring device to a meter unit must be guaranteed. This is typically achieved by using calibrated laser sources or molecular absorption lines as reference. With the redefinition of the SI unit kilogram, now defined in terms of Planck's constant h, new ways to measure mass and force with a direct reference to this constant are being actively explored. In addition to the Kibble balance implementations [6] made by numerous national metrology institutes over the years to realize the unit of mass, various mass, force and torque measurement instruments based on the Kibble principle were recently developed that offer high measurement accuracy and versatility. Tabletop Kibble balances [7,8] were designed to directly realize mass at the gram-level range with uncertainties on the order of 10 −6 . Also, multicomponent force and torques measurement systems [9], due to the use of the Kibble principle, allow force and torque calibration in all directions independent of the local gravitational acceleration with uncertainty levels of 10 −4 . In such measurement systems, the laser interferometer is the primary tool for traceable velocity measurement during the Kibble calibration, where the velocity needs to be determined down to parts-per-million (ppm) or in some cases even sub-ppm levels of uncertainty.
The conventional interferometer systems used for Kibble calibration typically employ helium-neon lasers, which offer a well defined emission frequency and exceptional frequency stability on the order of 10 −9 [10] together with very good coherence properties but are large in size and have a limited lifetime. These limitations can make the use of helium-neon lasers problematic in industrial applications. Furthermore, the optical design of a conventional interferometer system can be complex [1], requiring precise alignment and making the interferometer measuring head larger, which can impact the ability to integrate the measuring head into compact setups. These systems also suffer from periodic errors due to non-ideal alignment and beam leakage [11][12][13][14].
With the advancements in telecommunications technology, highly coherent laser diodes are becoming more widely available with coherence characteristics approaching those of helium-neon lasers. These laser diodes are starting to replace conventional laser sources in interferometer setups due to their extended lifetime, high output power and ability to directly modulate the laser frequency by modulating the laser injection current. In particular, the ability for direct laser frequency modulation enables the use of various pseudo-heterodyne phase demodulation techniques, eliminating the need for a complex optical interferometer setup, since the second optical quadrature signal for the phase demodulation does not need to be acquired. This has led to the development of compact interferometer measuring heads that consist of only a fibercoupled collimator [15,16], or even just using a ultra-compact GRIN-lens collimator [17], pointing at the measurement surface, offering new possibilities for compact integration potential and potentially low cost.
With these sensors, the displacement of the target surface can be extracted from the interference signal using various pseudo-heterodyne modulation techniques [15,[18][19][20][21][22][23]. The interferometric sensor proposed in this study employs a fibercoupled collimator as the sensor measurement head, in conjunction with a novel range-resolved interferometry method (RRI) [21] for phase demodulation. This sensor is denoted as fiber interferometric sensor (FIS) throughout this paper.
Unlike other pseudo-heterodyne modulation techniques methods, RRI enables the separation and simultaneous demodulation [24,25] of various interference sources within a single photodetector signal based on the respective optical path differences (OPDs) of the constituent interferometers. Even if the readout of only a single interferometric signal is required by the application, this approach reduces errors associated with multiple reflections and parasitic interference sources in the interferometer setup, leading to highly linear phase demodulation.
Pseudo-heterodyne fiber-coupled displacement sensing methods discussed above utilize highly coherent lasers with weak modulation. They have to be distinguished from other fiber-coupled displacement sensing concepts that use spectral evaluation in combination with low-coherence broadband or widely tunable sources. Such devices are often referred to as extrinsic Fabry-Perot interferometer (EFPI) sensors [26,27] and typically operate at stand-off distances at or below the millimeter-range. While EFPI sensors have the ability for absolute distance measurements due to the spectral evaluation used and can be deployed in very compact settings, they also lead to high phase noise levels due to the lower coherence of the light sources used [28]. Also, laser systems or spectrometric detection systems are typically more complex and expensive than the telecoms laser diodes and photo detectors used in this study. Furthermore, for the applications targeted in this paper, the principal ability for metrologically traceable wavelength calibration is crucial and broadband interrogation approaches used in EFPI-sensors are typically very difficult to calibrate.
The FIS proposed in this study enables traditional interferometer relative phase measurements with sub-nanometer precision, working over a broad stand-off distance that ranges from tens of millimeters to a meter. By calibrating the laser wavelength using one of the absorption lines of a hydrogen cyanid (HCN) gas cell, SI-traceable displacement measurements are in principle possible.
In this paper the performance of the FIS based on the RRI technique was characterized through direct comparison with a commercial He-Ne laser-based interferometer in tabletop Planck-Balance setup [7]. This evaluation is primarily aimed at examining the potential application of the sensor in force measurement instruments, with a focus on Kibble balance systems, but also highlights the potential of the RRI sensor for use as general-purpose vibration and displacement sensor.

FIS using RRI
The basic setup of the FIS is depicted in figure 1. The laser diode emits light that passes through a circulator to a fibercoupled collimator. Approximately 4% of the light is reflected from the flat-polished fiber tip back to the fiber due to Fresnel reflection, forming the reference beam. The rest of the light passes through the collimator lens, reflects off the target mirror, and is subsequently recoupled into the fiber. After being guided via a fiber-optic circulator onto a photodetector, the reflections from the fiber tip and the target surface create an interference signal that is subsequently evaluated.
The RRI method employs sinusoidal modulation of the laser's optical frequency. For precise demodulation, it is crucial to ensure that the modulation is sample synchronized with the sampling process. This synchronization is accomplished by utilizing a single data acquisition unit that shares a common trigger for both the digital-to-analog converter responsible for modulation output and the analog-to-digital converter used for sampling the photodetector signal.
A comprehensive explanation of the RRI method can be found in [21]. The following provides a concise overview of the method and the adaptations made for this study. The photodetector signal, which in general represents the superposition of interference signals from more than one constituent interferometer, can be expressed in simplified form as: where I 0 is the average intensity of the photodetector signal, V i is the fringe visibility of the ith interference, m i is the modulation index, also designated as the phase carrier amplitude in [21], and represents the amplitude of the phase modulation waveform resulting from the applied optical frequency modulation for a given OPD. It is calculated as: where ∆f is the optical frequency modulation amplitude, L i n is the OPD between the arms of the ith interferometer, and c is the speed of light. The demodulation process, illustrated in figure 2, involves multiplying the incoming photodetector signal by a time-variant complex carrier function, which is chosen according to the modulation index of an interferometer, and a window function to reduce crosstalk from other interferometers in the photodetector signal. It should be noted that the interference signal can be evaluated using the RRI method when its modulation index m > 20 rad. Consequently, adjustments must be made to either the modulation excursion or the OPD of the targeted interferometer to satisfy this criterion. The main difference between the RRI method and other pseudo-heterodyne phase demodulation techniques is that it uses a sinusoidal optical frequency modulation with a much larger wavelength excursion (up to 0.5 nm). The large wavelength excursion results in a larger modulation index, allowing to separate and individually demodulate signals from multiple interference sources based on their OPD. While a larger wavelength excursion initially has the drawback of requiring a larger detection bandwidth and places more demand on the modulation capabilities of the laser diode, the key advantage of this approach is the gain in flexibility. In RRI the phases of multiple interferometers can be interrogated simultaneously if their modulation indices vary sufficiently, typically ≈20 rad or larger [24]. Even if the interrogation of only a single interferometric signal is desired, the ability to accurately suppress signal contributions due to parasitic or multiple reflections that are physically present in most optical setups allows the use of simplified and very compact optics and high alignment tolerances, while maintaining a very good linearity performance.

Planck Balance velocity mode measurement
Recently developed Planck-Balance 2 system [7] uses the Kibble balance principle to compare the mechanical force applied to the system to the electromagnetic force on a currentcarrying coil in a magnetic field. The velocity mode of the balance determines the electromagnetic force factor (Bl-constant) of the coil-magnet system, which is then used in the force mode to calculate the applied force.
During the velocity mode, the coil oscillates up and down in the magnetic field, and its position z is measured using an interferometer over several complete cycles, while the induced voltage U is measured using a digital multimeter. The Blconstant is then calculated using the equation: where U 0 is the amplitude of the induced voltage, ω is the frequency of the coil displacement, and z 0 is the amplitude of the coil displacement. These parameters are usually determined through a three-parameter or multi-harmonic sine fit algorithm [29] applied to the displacement signal z from the interferometer. The uncertainty of the coil displacement frequency ω is primarily influenced by the accuracy of the sample timing and can be determined with a relative uncertainty of 10 −8 level by using a GPS reference as the clock source to trigger the measurements. However, the uncertainty of the amplitude of the coil displacement z 0 is influenced by various factors such as wavelength accuracy, environmental conditions, Abbe-error and interferometer cyclic errors. This work will thoroughly examine the impact of these factors on the measurement uncertainty.

Experimental setup
The FIS was integrated into the existing Planck-Balance setup [7] as shown in figure 3. The sensor and the commercial interferometer were positioned in such a way that they both point towards the same mirror, but from opposite sides. This configuration enables a direct comparison of the displacement measurements taken by the sensor and the commercial interferometer. The commercial homodyne differential interferometer (DI) operates at a wavelength of 632.991 2352 nm with laser wavelength stability better than 2 × 10 −8 , offering displacement measuring resolution of 5 pm. The combined relative measurement uncertainty of the commercial interferometer relevant for this comparison is 0.78 ppm. The corresponding uncertainty contributions are included in table 1.
The left-hand side of figure 3 illustrates the mechanical configuration of the experiment. To account for drifts in the air refractive index and wavelength, a commercial DI employs a stationary second mirror as a reference to minimize interferometer dead path, therefore reducing drifts. To achieve the same effect with the FIS, a second sensor was installed and aimed at a local reference mirror. During post-processing, the reference signal was subtracted from the target signal to compensate for any drifts. It is worth noting that it is possible to acquire the reference mirror position using only one FIS by splitting the output beam and directing it to both mirrors, as previously demonstrated in [24]. However, this setup would result in different interferometer OPDs, thereby losing the ability to fully compensate drifts and adding complexity to the optical setup. Thus, we opted for a more straightforward approach, utilizing two separate FISs of nominally equal OPD.
A multichannel data acquisition unit dSPACE MicroLabBox was utilized to simultaneously acquire data from two FISs and a commercial DI. The accuracy of the sampling time was guaranteed with an uncertainty of 1 × 10 −8 s −1 by using a GPS reference clock connected to the Keysight 33500B signal generator, which generates the trigger signal for the measurement. The GPS reference was also connected to the reference clock of the signal generator, which generated the modulation signal to modulate the laser injection current and, thus, the laser wavelength, ensuring synchrony between the sampling and modulation, which is required in the RRI method.
The modulation frequency should be high enough to avoid introducing 1/f noise to the demodulated signal, leading to high sampling frequency requirements. However, in the current setup, due to the need to acquire multiple channels simultaneously and limitations of the used DAQ system, the maximum available sampling rate was 100 kHz. Therefore, the modulation frequency, and thus the resultant phase data rate, was chosen to be 500 Hz, much lower than the many tens of kHz that have previously been achieved using RRI [24,30].
For the laser source, an Elana Photonics EP1550-NLW-B-100 laser diode, operating at a nominal wavelength of 1550 nm with a narrow linewidth of 100 kHz was utilized in the experiment. Laser wavelength was modulated with a wavelength excursion of about 0.3 nm. 50% of the laser output was sent through a HCN gas cell to a photodetector. The signal from the photodetector was used to manually set the laser wavelength to the peak of the gas absorption line as detailed later. The remaining light from the laser was equally split for the two FIS. For each FIS, the laser light was directed to the fiber collimator, serving as the measuring head of an interferometer, via a fiber optic circulator, and the reflected light was then sent back to the photodetector by the circulator. A climate sensor was used to acquire environmental parameters (temperature, air pressure and humidity), and, as detailed later, this was used to apply a correction for the air refractive index for each interferometer signal.

Results and discussion
In this section the uncertainty budget for FIS is evaluated first. Then, a comparison between the measurement with the FIS and the commercial DI, which was used as a reference displacement sensor, is presented.

Uncertainty budget of velocity measurement
The uncertainty budget for velocity measurement using a FIS was evaluated based on the primary influencing parameters and is detailed in table 1. These calculations correspond to a measurement in which the balance coil underwent sinusoidal displacement with an approximate amplitude of 20 µm and a frequency of 0.25 Hz. The uncertainties are divided into two parts: one caused by the FIS and used hardware, and the other part represents the uncertainties of the commercial interferometer that are relevant for direct comparison between two interferometer systems, and with the total uncertainty for the whole comparison also calculated and stated.

Laser wavelength.
In this study, a fiber-coupled HCN molecular absorption gas cell is utilized as a wavelength reference to set the working point of the laser to a gas absorption line at 1549.730 51 nm, which is known with an uncertainty of 0.04 pm or 0.026 ppm [31]. Figure 4 illustrates the relationship between amplitude response and wavelength, acquired by sweeping the laser temperature, thus altering the laser wavelength, and measuring the output amplitude at the gas cell. The temperature was subsequently recalculated into wavelength change using the temperature tuning coefficient  (3) is mechanically coupled to the coil (6). The two beams of the commercial differential interferometer (DI) are shown with red arrows (5) and directed to the measuring and reference mirrors similarly to the FIS, but from the opposite side. On the right side of the figure, the structure of the experimental setup is depicted. of the laser diode. The determined full-width half-maximum width of the observed absorption line is 14 pm, which closely corresponds to the theoretical value of 16 pm provided in the datasheet. By adjusting the laser temperature and offset injection current to the point of minimal amplitude before measurements, it was ensured that the laser wavelength was initially centered on the line within a range of ±1 pm.
By employing active stabilization with the fringe lock-in method, it should become possible to lock the laser wavelength to a gas absorption line with a relative wavelength uncertainty of typically 3 × 10 −8 [15]. However, the wavelength modulation excursion necessary for this should be ideally much less than the width of the gas line, rendering it incompatible with the large modulation excursion required for the RRI-method. This will require specialized laser stabilisation techniques to address this issue, and their development will be the focus of future research efforts.
Furthermore, when a large wavelength modulation is applied to the laser, a center wavelength shift usually occurs, contributing to the measurement uncertainty as well. For the modulation parameters used in the experiment, the center wavelength shift is estimated to be no more than 3 pm. Also, the non-linearity of the injection current modulation process can cause shifts in the effective demodulation wavelength, which is another factor contributing to the wavelength uncertainty for widely modulated laser diodes. However, the modulation waveform harmonics can be determined from the RRI carrier waveform [21] and these values used in simulations to estimate the wavelength shift. Simulating the interference signal with the acquired modulation waveform parameters for this experiment, the deviation of the effective wavelength was estimated to be no more than 0.5 pm.
Both the center wavelength shift and effective demodulation wavelength represent systematic errors that can be corrected through detailed examination and calibration. However, in the scope of this study, no calibration has been performed, and the estimated parameters are considered as worst-case wavelength uncertainties. Future research will focus on a more comprehensive assessment of these factors or use real-time control to eliminate them. Therefore, the combined relative uncertainty for the three factors affecting the wavelength uncertainty is estimated to be 2.1 ppm, which is currently the dominating uncertainty source (see table 1).

Beam orthogonality.
The mutual geometric alignment of the two interferometers is of crucial importance for their comparison. The commercial interferometer has been carefully align to the mirror as detailed in prior work [7]. For the FIS, comfortable live alignment is possible by fast computation of the RRI range view that is illustrated in figure 5(b). The range view displays the dependence of the demodulation signal amplitude on the demodulation phase carrier amplitude, which corresponds to the OPD of the constituent interferometers present. The primary signal is represented by a large peak at approximately 50 mm, whereas the signal at 100 mm results from the beam double-passing the space between the fiber and mirror due to multiple reflections. In conventional pseudo-heterodyne phase demodulation techniques, this could cause significant periodic non-linearity in the signal. However, the RRI method enables the main signal to be demodulated independently, without being affected by unwanted interference signals, while the double-pass signal is simply ignored. The method used to create the range view and to estimate the OPD of an interferometer has previously been described in detail [32]. The range view provides a means to ensure proper alignment of the sensor head to the single pass mode [15], which occurs for perfect perpendicular alignment between the optical axis and the target mirror, and distinguish it from the double pass mode [33], which occurs when the beam double-passes the space between the fiber and mirror due to slight misalignment between the sensor head and target. To align the sensor beam orthogonal to the target, it is thus necessary to align the sensor to the maximum amplitude of the main peak. To verify the quality of the alignment achievable using this method an additional setup outside the Planck-Balance is used. Here, the same collimator as in the FIS (Thorlabs F240FC-1550 with an aspheric lens having a focal length of 8.18 mm) and a commercial autocollimator are arranged to point at the same mirror, where the angular alignment of the mirror is adjustable. Using the data from the autocollimator as independently measured angle reference it is possible to estimate the dependence on the mirror alignment of the peak height in the range view, with results shown in figure 6. Here, using manual kinematic mounts identical to those used in the main setup, a reasonable threshold for peak alignment of 95% of its maximal possible value can be reliably reproduced, thereby ensuring beam orthogonality to the target mirror within 120 µrad. Therefore, this value is taken as the uncertainty of the beam orthogonality in table 1.

Environmental factors.
Since the coil movement is measured in air, an accurate estimation of air refractive index is needed. For that, environmental parameters were acquired with climate sensor and air refractive index was calculated using modified Edlén equation [34], with separate corrections applied to both FIS and DI due to the different wavelengths used. The measurement uncertainties of the environmental sensors and their contributions into uncertainty of the air refractive index are listed in [7]. The combined relative uncertainty of the air refractive index for this under laboratory conditions was determined to be below 0.14 ppm.

Periodic nonlinearities.
Nonlinearity errors in interferometry [35], i.e. periodic deviations from the true measurement value, can affect the accuracy of determination of the sinusoidal movement amplitude for the Kibble calibration, depending both on the exact amplitude and the exact interferometric phase at the start of the sinusoidal movement. As observed in figures 7(b) and (c), the deviations of the the quadrature interferometer signals from the ideal circle indicate the presence of nonlinearity error. To correct periodic errors, a Heydemann correction [11] is applied to the obtained quadrature photodetector signals. However, this method cannot correct any high-order nonlinearities.
The measurement results by both the FIS and the commercial interferometer are analysed for nonlinearity errors. Here it should be noted that in this work, to ensure exact synchronicity, the sine and cosine analogue output signals of the the commercial interferometer are directly sampled by the data acquisition unit and therefore the internal nonlinearity correction of the commercial interferometer system is intentionally bypassed. Figure 7(a) presents the residuals acquired by subtracting the multi-harmonic sine fit of the displacement signal from the displacement signals itself for both interferometers. Here, the top panel of figure 7(a) shows example signals before and the bottom panel of figure 7(a) after the Heydemann correction has been applied to both interferometer signals. Residuals are displayed at the turning point of the sine displacement for better visualization of the cyclic error and this dataset is also used for the subsequent calculation of the Lissajous figures. In the top panel of figure 7(a) it can be seen that the DI residual signal before correction exhibits a distinct periodic error pattern with an amplitude of approximately 1 nm, while the FIS residual signal has no apparent periodic error and the residual is mostly composed of what is assumed to be mechanical and phase noise of standard deviation of about 0.16 nm. In the bottom panel of figure 7(a), after the Heydemann correction, both signals show no directly visible cyclic errors.
To further analyse the nonlinearity errors the Lissajous figures are plotted. Here, figure 7(b) shows the Lissajous figures without Heydemann correction at full scale, while figure 7(c) plots an enlarged version of the same data. It can be seen that the DI exhibits both elliptical and higherorder nonlinearities, while the FIS exhibits an almost circular shape. Figure 7(d) then compares the zoomed in Lissajous figures after the Heydemann correction on the same scale as figure 7(c). Here it can be seen that for the DI the elliptical nonlinearities have disappeared but higher order nonlinearities remain, similar to those observed by Dai and Hu [36]. The amplitude of the maximum phase errors of the interferometer nonlinearities are also estimated using the method described in [37]. Here without correction the maximum phase errors are 1.2 nm for the DI and 50 pm for the FIS, while after Heydemann correction the resulting maximum phase errors are 50 pm for the DI and 10 pm for the FIS. This confirms the very high linearity performance of RRI-based signal processing that was also previously observed [17]. While the experiments shown here give an indication of the linearity performance achievable with both interferometer type, more detailed future analysis, for example following the method of Köning et al [35], is needed to fully compare and assess the performance but this is beyond the scope of this paper.
The impact of interferometric cyclic errors on the final estimation of sinusoidal movement amplitude has also been explored through simulations. The size of the sine-fit error is an oscillating function of the displacement amplitude range, with the amplitude of the oscillation decreasing with increasing range. The simulations reveal that for a movement amplitude of 20 micrometers the worst-case error in the resulting amplitude estimation is approximately 0.1 ppm when the cyclic error is at the 10 pm level as for the FIS and 0.5 ppm for cyclic errors at the 50 pm levels as for the DI. Therefore, for both interferometer types, the observed nonlinearity errors at the levels seen after Heydemann corrections are relatively small and currently not dominant, however, for smaller movement amplitudes the impact will be correspondingly larger. It is important to note that the quadrature signals from the commercial differential interferometer were directly acquired from the raw photodetector signal and not obtained from the output of the interferometer processing unit in order to achieve better synchronization with the FIS output.

Errors due to non-ideal motion of the load carrier.
For the Planck-Balance velocity mode measurement, it is also essential to understand how the measured movement of the mirror corresponds to the real movement of the coil. Since these uncertainties depend on the setup construction and not on the sensor itself, they are not included in table 1. The angular misalignment of the mirror normal and the motion axis leads to a cosine error that was analysed in [7]. Here, a second interferometer was used to measure the orthogonal displacement of the side of the mirror cuboid, thereby measuring the horizontal displacement and determining the cosine error. The angular misalignment is estimated to be 3.5 mrad with an uncertainty of 450 µrad, and can be corrected during data analysis.
Furthermore, accurate measurement of the tilt motions of the load carrier, which contribute to the Abbe error, was conducted using an additional 3-beam interferometer [7]. This allows for the correction of the Abbe error, with a combined measurement uncertainty of 80 pm for the case of a motion amplitude of 20 µm and frequency of 0.25 Hz.

Measurement results
The main coil of the balance was displaced sinusoidally with an approximate amplitude of 20 µm and a frequency of 0.25 Hz. Figure 8(a) shows two periods of the displacement signal, while figure 8(b) shows the difference between the displacement signals acquired with FIS and DI.
Ten measurements were conducted where the position of the coil was simultaneously acquired for 60 s for each of 10 measurements using both the commercial interferometer and the FIS. The amplitude and frequency of the displacement signal were estimated using a multi-harmonic sine fit for each measurement. The velocity amplitude was calculated for both FIS and DI using the estimated values. Figure 9 shows the relative deviation between the velocity amplitudes measured with FIS and DI for each of the ten measurements. The mean value of velocity amplitude for FIS and DI is 31.282 31 µm s −1 and 31.282 37 µm s −1 , respectively, with a mean relative deviation of −1.9 ppm between them. The deviation values obtained are consistent with the theoretical considerations that were  described previously and are summarised in table 1. In the current experiment, the deviation between the interferometers is mostly caused by the wavelength uncertainty of the FIS; therefore, the superior level of periodic nonlinearity of FIS observed in section 4.1.4 does not manifest itself yet in the final comparative result. However, following future progress on minimizing the wavelength uncertainty, the differences between the two interferometric approaches in terms of periodic nonlinearity are expected to become the dominant sources of uncertainty. Furthermore, if smaller sinusoidal motion amplitudes are applied, the relative influence of periodic nonlinearities becomes correspondingly larger, therefore the observed improvements in terms of periodic nonlinearity performance are very important.

Conclusion
In this study, a FIS has been qualified for measuring the velocity amplitude of a sinusoidally vibrating object, with a target application of measuring the velocity of the moving coil during the velocity mode of the Kibble balance. The velocity amplitude of the coil, which is oscillating with a 20 µm amplitude and a 0.25 Hz frequency, was measured with the FIS and is 31.282 31 µm s −1 with a relative deviation of −1.9 ppm compared to a commercial differential interferometer, indicating good agreement between the two sensors. An uncertainty budget for the comparison between the FIS and the commercial interferometer has been presented, which considers various factors that contribute to the uncertainty of the velocity measurement, including the stability of the laser wavelength, the refractive index of air, periodic nonlinearities in the interferometer, and errors due to environmental factors and setup configuration. By estimating and considering these factors, the relative uncertainty of the velocity measurement was determined to be 2.1 ppm. The dominant contribution originates from the wavelength uncertainty, which can be improved in future work through calibration using a known wavelength reference and by actively locking onto the gas absorption line, expecting to obtain at least an order of magnitude improvement. Overall, this study demonstrates the potential of FISs for high-precision displacement and velocity measurements and highlights its very high linearity performance, with promising applications in fields such as precision manufacturing, nanotechnology and metrology. Due to the compactness of the sensor, coupled with the high performance that has been demonstrated, the sensor with multichannel measurement configuration will now to be integrated into a multicomponent force and torque measurement system for traceable velocity mode measurement during the self calibration process using Kibble Balance principle.

Data availability statement
The data that support the findings of this study will be openly available following an embargo at the following URL/DOI: https://doi.org/10.5281/zenodo.8060258. Data will be available from 30 August 2023.