## Abstract

Frequency scanning interferometry (FSI) is a promising technique for absolute distance measurement and has been demonstrated in many industrial applications. However, in practice, the measurement precision is limited and sensitive to the variations of the measured distance while sweeping the optical frequency of the laser. The induced errors would be amplified by hundreds of times due to the magnification effect. In this paper, an incremental interferometer was established on the basic scheme of the FSI system for monitoring the variations of distance. The compensation could be achieved by multiplying the heterodyne signals from monitor and measurement interferometer without complex and time–costing data processing. The system performance has been verified by experiments for different kinds of vibrating targets. Finally, after compensation by suppression of the magnification effects, a measurement precision of 4.26 *μ*m has been achieved in a range of 10 m.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

High–precision distance measurement is of significant interest and indispensability, especially in the manufacturing industry and big science projects [1,2], where the relative accuracy is required as around $10^{-6}$ in a range of tens of meters [3,4]. In general, the operating environment in the open air of large–scale volume is the most challenging for the promising precision as expectation. The environmental changes, such as mechanical vibration, airflow turbulence, and gradient of refractive index, would lead an unexpected and limited precision [1,5,6].

Since 1980s, the frequency scanning interferometry (FSI) has brought more attention to the field of the industry because of its excellent precision, flexibility, and traceability [7–9]. Unlike the incremental interferometer, the absolute distance measurement could be realized with FSI by tuning the laser frequency instead of moving the target continuously [7,9,10]. Another approaches for absolute distance measurement based on optical combs and multi–wavelength laser sources are also broadcast widely, but they seem complex and luxurious for industrial applications [11,12]. Taking advantage of time–of–flight (TOF) and interference methods, the FSI has been demonstrated on a scale from millimeters to kilometers with applications of optical frequency domain reflectometry (OFDR) [13], optical coherence tomography (OCT) [14] and frequency-modulated CW laser detection and ranging (FMCW LiDAR) [15–17], for example.

However, in practice, the ideal precision benefits from the careful control of the potential error sources in FSI [1]. The variation of measured optical path difference (OPD) would lead merciless loss on precision when the FSI was implemented on dynamic targets or in uncontrolled environments [1,18]. The “dynamic errors” would be magnified by dozens or even hundreds of times [16], which could be called the magnification effect in FSI. Although the impact could be reduced by increasing the tuning range of frequency or decreasing the measurement time, it can not be fully eliminated [10]. Dual sweep FSI with a second tunable laser seems an ideal way to solve this problem by sweeping in an opposite direction simultaneously [16]. However, complex control methods for promising the lasers sweeping simultaneously and the increasing costs due to the second tunable laser need to be considered. The alternative principle has already been reported by J.J. Martinez et al. in 2015: a second swept laser, which is perfectly synchronized with the origin, was generated by the nonlinear effect with four-wave mixing (FWM), but it also cannot avoid complex systems [19]. As another simple and effective method, the variation of the measured path could be monitored by a second incremental interferometer [20–22]. However, the two interferometers must be arranged along the same optical path for reducing the introduction of additional errors. In addition, the point by point compensation process limited the efficiency of measurements.

In this paper, the modified model of the FSI has been derived for analyzing the magnification effect. In particular, the dynamic errors for distance measurement has been emphatically introduced based on the mathematical model. Then, an optimization method based on a heterodyne scheme has been proposed for suppressing the magnification effect of dynamic errors in FSI. The heterodyne interferometer is combined by a wavelength multiplexer (WDM) to monitoring the variation of OPD, which is for enhancing the robustness of the system in harsh environments. Besides, the compensation of dynamic errors has been done in real–time. Furthermore, the proposed system is evaluated in this paper by dynamic measurements with different vibrations. Finally, a standoff absolute distance of 10 m has been measured by the proposed system, and the precision is better than 4.26 $\mu$m corresponding a relative order of $10^{-7}$.

The paper is organized as follows. In Section 2, the mathematical model of the FSI and simulations are established for analyzing the dynamic errors. The principle of compensated strategy has been proposed in Section 3. In Section 4, the experimental verification is carried out with different experiments. Finally, the results and prospects are concluded in Section 5.

## 2. Mathematical descriptions of FSI for dynamic targets

#### 2.1 Overview of the basic principle in FSI

In general, the typical FSI system is designed by the modified conventional interferometer. For a certain optical path length difference $D$, the absolute interferometric phase at instantaneous laser frequency of $\nu (t)$ could be expressed as

where $c$ represents the speed of light in vacuum and $\phi _0$ is the initial phase. When the laser frequency sweeps from $\nu (t_0)$ to $\nu (t)$ continuously and monotonously, the measured distance could be calculated byAn overview schematic of typical FSI is shown in Fig. 1, which consists of two interferometers: a measurement interferometer with an unknown OPD $D_{m}$ and an auxiliary reference interferometer with calibrated OPD $D_{r}$, both sharing a common frequency tunable laser source. The exact knowledge of $D_r$ is realized by inverting the measurement process of the system to a calibrated measurement length. According to Eq. (2), continuous phase changes could be observed in both of the interferometers when the laser frequency varying in a well-defined range of $\Delta \nu (t)$, which could be expressed as:

Therefore, the measured optical path could be determined by: Conclusively, the measured length could be obtained by:Practically, the measured length could be determined by measuring the ratio of phase variations in measurement and reference interferometer [9,10] or the frequency of re–sampled interferometric signal [16,24]. Frequency measurement based FSI techniques shows more flexibility and robustness, which are preferred in industrial scenarios, especially for applications with low reflectivity [19]. However, there is a conflict between the data processing time and measurement precision. Millions of data would be acquired and analyzed by fast Fourier transform (FFT), which severely limits the measurement speed. Phase information based FSI techniques is more commonly used in high precision measurements with a single target. Nevertheless, stochastic noise would lead a precision loss to the measurement result, makes it sensitive to the signal to noise ratio (SNR) of the interferometric signal. To mitigate the influence of stochastic noise, linear regression could be used for extracting the ratio of phases [20]. Furthermore, the residuals after regression would be helpful for further study of other contributions on measurement errors.

#### 2.2 Magnification effect of dynamic errors

It is worth to mention that the derivation above base on an assumption that the OPD of measurement interferometer is stable during the sweep. Practically, the experiments of FSI are normally implemented in uncontrolled environments. Mechanical vibrations, air turbulence, thermal effects, and other environmental factors will affect the OPD, lead to significant precision losses. For a target with dynamic OPD of $D_m(t)$, the phase change of measurement interferometer could be rewritten as:

According to the analysis above, the optical path length variation during the sweep will lead severe precision loss to the measurement result. The magnified errors introduced by the drifts could be cancelled by dual sweep laser system efficiently. A contrary magnified drift error $-{v(t)\Delta {D_m}}/{\Delta \nu (t)}$ could be obtained by tuning the laser in opposite direction. However, the cost and complexity of the system increase significantly after introducing an additional tunable laser. The problem is overcome by an ingenious method proposed by NPL [19]. A mirror swept light is generated by using degenerative four wave mixing with the help of a fixed frequency laser. Nevertheless, the complexity of the system and the low SNR of the interferometric signal need to be overcome. Another effective approach is subtracting the drift introduced phase changes from the measured phase directly. It could be realized by establishing an incremental interferometer in the measured path by introducing a single frequency laser to the measurement interferometer. The combined light is distinguished in frequency domain, which is achieved by shifting the swept light and fixed light with different frequencies, respectively. However, the measurement efficiency is decreased by sophisticated data processing. Furthermore, the path length difference between the measurement interferometer and the monitor interferometer will lead unexpected errors to the measurement result.

## 3. Suppression of the magnification effect in FSI

An improved FSI system for suppressing the magnification effect of dynamic errors in FSI is proposed in this section, which is shown in Fig. 3. The measurement interferometer is established in heterodyne scheme. A continue–wave (CW) single frequency laser is introduced to the measurement interferometer to monitor the optical path variation during the sweep. The wavelength of the single frequency laser will not overlapp with the sweeping light. Different from the former researches, the sweep and fixed light combined together and shared a common optical path to avoid the introduction of additional errors. The combined light is separated at a wavelength division multiplexer (WDM). The frequency–shifted interferometric signals are obtained at PD2 and PD3, respectively. According to Eqs. (1) and (6), the interferometric phases of FSI and monitor interferometer could be shown in:

It could be inferred from Eq. (8) that the interferometric phases of FSI combines the phase caused by tuning the optical frequency of laser, movement of the target and acoustic optical modulator (AOM). The first term of both equations are the same and the final term, which caused by target’s movements will be the same when the frequency of the fixed light equals to the stop frequency of the swept light. The phase difference between the two signals will just be the phase changes caused by sweeping the laser’s frequency. A mixer (Mixer2) is used here to get the product of the two signals. The output signal (Signal3) of the mixer could be expressed in

## 4. Experiments and results

In Section 3, method for suppressing the magnification effect of dynamic errors in FSI system has been proposed. In order to evaluate the performance of the system discussed above, experiments were implemented in different situations.

#### 4.1 Experimental set–up

An improved FSI absolute distance measurement system has been established according to Fig. 3. An external cavity laser diode (81606A #116 series, Keysight) is used as the swept laser source. It could provide a wide tuning range covers from 1490 nm to 1640 nm and tuning speed up to 200 nm/s. A fixed frequency laser (Orion, RIO) with an output power of 20 dBm and 1550.12 nm wavelength is used for monitoring target movements. The swept laser source sweeps from 1522 nm – 1537 nm to distinguish from the fixed light in WDM. The light emitted from the swept laser is divided into two parts by a 90/10 fiber coupler: 10% of the light is used for illuminating the auxiliary interferometer and optical frequency monitoring while the 90% part combines with the fixed frequency components to perform an absolute distance measurement. The auxiliary reference interferometer is established in Mach-Zender design with a fiber delay line about 100 m , which is equivalent to an optical path difference of 144.886517 m (calibrated by a commercial interferometer). However, the temperature variations and mechanical vibrations would affect the OPD of auxiliary reference interferometer severely. Therefore, a thermal isolation isolator is necessary here to keep the auxiliary interferometer away from the influence of the environmental effects. Monitored by an incremental interferometer, the relative stability of auxiliary interferometer is better than $10^{-7}$ in 1 hour, whose influence on the measurement results could be ignored. In measurement interferometer, the combined light go passed another fiber coupler with a beam–splitting ratio of 75/25. The 75% part passes through a circulator and emitted from a collimator towards the target. The reflection light goes back again along the same path, combine with the 25% part again after frequency shifted by AOM (T-M080-0.4C2J-3-F2S, Gooch & Housego). The operating frequency of AOM is set to 80MHz. The interference lights of auxiliary and measurement interferometer are detected by photodetectors PD1, PD2, and PD3 (PDB470C, Thorlabs), respectively. The auxiliary interferometric signal detected by PD1 could be expressed as Signal1, which is used for monitoring the laser frequency. The signals from monitor and measurement interferometer, which are detected by PD2 and PD3 respectively, mixed together at Mixer2, and filtered by a low passed filter (LP Filter2). Then the FSI measurement signal after compensation could be acquired, which is expressed as Signal3. Original FSI signal without compensation was also acquired in the same time for comparison. By mixing the signal of PD2 and the local oscillator signal of AOM at Mixer1, the uncompensated FSI measurement signal could be acquired after removing the high frequency components, which is expressed as Signal2. All the interferometric signals are digitized and recorded by an oscilloscope (DPO5104B, Tektronics) at a sampling rate of 20 MS/s. The phase changes would be extracted from the interferometric signal using a Hilbert transform on the computer.

#### 4.2 Data processing

The data processing method of the digitized interferometric signal was shown in Fig. 4. Affected by the AOM, the measurement signal was modulated by a low frequency noise, shown in Fig. 4(a). In Fig. 4(b), the instantaneous phase was obtained by applying Hilbert transformation on both of the measurement and reference signals. After unwrapping the wrapped phases, a series of data about the phase–time variation in the measurement interferometer could be obtained, same for the reference interferometer, shown in Fig. 4(c). Linear regression is used here for calculating the ratio $\Delta {{\phi }_{m}}(t)/\Delta {{\phi }_{r}}(t)$. The final linear regression equation is shown in Fig. 4(d).

Figure 5 shows the comparison of the phase residuals and distance spectrum for a dynamic target. The regression residuals before compensation show a high degree of correlation with the actual variation of OPD, which is shown in Fig. 5(a). However, the linear drift of the target could not be eliminated by linear regression, which would introduce a deviation to the measurement result. After the compensation by monitor signal, the regression residuals reduced from $\pm$3 rad to $\pm$2 rad, which is shown in Fig. 5(b). Meanwhile, a second–order component should be noticed in both of the residuals before and after compensation, which is most probably caused by the dispersion mismatch effect. The theoretic curve of the second–order dispersion effect was plotted according to the experimental parameters. The previous deduction was proved by the good consistency of the simulation and the actual residuals. Detailed description of the dispersion mismatch effect could be found in [25]. The same conclusion could be obtained from the spectrum of the re–sampled signals, which is shown in Fig. 6. The full width at half maximum (FWHM) of the distance spectrum before compensation was broadened to 805 $\mu$m by the dynamic measured length. After compensation, the spectrum has been narrowed to 215 $\mu$m. A new measurement interferometric signal was reconstructed according to the linear regression equation. Removing the influence of the dispersion mismatch effect, the spectrum of the new signal shows good accordance with the simulation signal. The FWHM of the new signal is close to the theoretical value, which demonstrates the validity of our method.

#### 4.3 System verification

Various OPD variations could be introduced by different kinds of environmental factors, such as mechanical vibrations, air fluctuation, and thermal effects, etc. According to Fig. 2(a), the movements of the target could be seen as a superposition of periodic motion and linear drift during the sweep. In this section, the system would be verified by applying different kinds of motions to the targets.

The experimental set–up is shown in Fig. 7. The proposed FSI system is used for measuring the distance to a flat mirror (PFD10–03–P01 , Thorlabs), which is attached to a piezo linear precision positioner (P–622.1CD, Physik Instrumente). The positioner is mounted in the direction of the FSI measuring light. The moving direction of the positioner is strictly aligned with the measurement beam. Controllable distance variations could be achieved by adjusting the voltage of the driving signal. As a result of the hysteresis effect of PZT, the actual movements of the mirror is not consistent with the driving signal. Different kinds of motion would be generated by adjusting the driving signal to evaluate system performance. The feedback signal of the PZT driver could be used as a reference of the motion of the target. Calibrated by the incremental interferometer, the displacement of the target is proportional to the feedback signal by about 50 $\mu$m/V. All the measurements were performed in both basic FSI and improved FSI for comparison.

Firstly, the system is applied to vibrating targets with different frequencies. Before the experiment, an incremental interferometer was used to monitor the vibration of a target in an uncontrolled environment. The measurement results indicate the frequency of the vibration was about 5 Hz to 10 Hz. Some studies show that the frequency of the mechanical vibrations could vary from 1 Hz to 100 Hz in industrial scenarios. Considering the dynamic performance of the positioner, the frequency of the driving signals is set from 10 Hz to 50 Hz. As a result of the hysteresis effect, the actual amplitudes of vibration were not the same at different frequencies. The amplitude of vibration is set to 8 $\mu$m for all the experiments by adjusting the driving signal according to the feedback signal. 10 times measurements were taken for each frequency. Figures 7(b) and 7(d) show the measurement residuals before and after compensation, respectively. The standard deviation (STD) is used here for evaluating system performance. Before the compensation(see Fig. 7(b)), the maximum STD was as large as 127 $\mu$m. Benefits from the compensation of the improved system, STD about 4 $\mu$m has been achieved, shown in Fig. 7(d). Then the system measured vibrating targets with different amplitudes and fixed frequency of 10 Hz. The amplitude of vibration varies from 7.5 $\mu$m to 37.5 $\mu$m. A similar conclusion could be got from the STD of measurement results, which is shown in Figs. 7(c) and 7(e). The proposed system suppressed the magnification effect of dynamic errors efficiently, shown strong robustness to the variation in the measured optical path.

#### 4.4 Precision evaluation

In this section, the proposed system was evaluated by conducting experiments in a volume without temperature control and vibration isolation, which is close to the industrial environment. The experimental implementation is shown in Fig. 8. Absolute distance measurement were presented in a range of 10 m with 1 m interval. 50 individual measurements was presented for each measurement point by the proposed system. The absolute distances between points were calculated according to the coordinates obtained by laser tracker (AT901, Leica), which is regarded as the reference. The retro–reflectors are used for coordinate measurements by laser tracker and act as the target for the FSI system at the same time. Position sensing detector (PDQ30C, Thorlabs) are introduced in order to promise all the measurement points were located on the same line strictly. The reflection lights from the target are separated into two parts by a beam splitter. Part of the light was combined with the light of the reference arm to perform an absolute distance measurement while the other part is reflected to the position sensing detector (PSD). The misalignment would be eliminated by adjusting the position of the target with a 3–axis stage according to the output voltage of PSD. Temperature and humidity of the measured optical path are recorded by environmental compensator (XC-80 , Renishaw) at the first measurement and the last measurement for each point.

The measurement residuals and standard deviation are shown in Figs. 9(a) and 9(b). The empirical uncertainty limits are estimated for both measurement results before and after compensation, which is presented as the black and red bar, respectively. The standard deviations of the measurement results decrease from 44.30 $\mu$m to 4.26 $\mu$m with compensation by the proposed system. However, the measurement accuracy compared with the laser tracker is not ideal with a maximum deviation of about 40 $\mu$m, which is most probably limited by the accuracy of coordinate measurements and the alignments of the measured points. In this experiment, the laser tracker was used to get the location of each measurement point. The reference length was calculated by the coordinates of these points. As the measurement uncertainty of the laser tracker is about 15 $\mu$m + 6 $\mu$m/m, the measurement residuals could not reflect the real measurement accuracy of the proposed system.

From Eq. (5), the measurement uncertainty is related to the uncertainties of the air refractive index of measurement interferometer $n_m$, the optical path of reference interferometer $D_r$ and the phase variation determination in measurement and reference interferometer $\Delta \phi _m(t)$, $\Delta \phi _r(t)$ respectively. The first term $n_m$ is mainly determined by the environmental temperature, humidity, and pressure, which are stable at 0.13 K, 11 pa, and 0.65% for each measurement point respectively. According to the Ciddor formula [26], the uncertainty of the air refractive index is about 0.13$\times 10^{-6}$. As the reference interferometer was kept in a thermal isolation isolator, the stability of the second term $D_r$ was less than 5 $\mu$m, which is equivalent to a relative uncertainty about 3.42$\times 10^{-8}$. The phase variations $\Delta \phi _m(t)$ and $\Delta \phi _r(t)$ will introduce a measurement standard uncertainty of 1.81 $\mu$m. Therefore, the combined measurement uncertainty of the system is 1.81 $\mu$m $\pm$ 0.13 $\mu$m/m.

## 5. Conclusion and future work

An improved FSI system was proposed in this paper for suppressing the magnification effects due to OPD variations during the measurement. The dynamic errors could be suppressed to a considerable level without complex data processing, which greatly improves the robustness and measurement efficiency of the FSI system. The validity of the system is proved by applying on different vibrating targets. Measurement precision of 4.26 $\mu$m was achieved within 10 m in uncontrolled environments. In future work, the system would be further optimized. Compensation of the dispersive effect and improvement of the phase detection system should be considered for higher accuracy. Moreover, the system would be used for large volume coordinate measurements with the help of fast steering mirror (FSM).

## Funding

National Natural Science Foundation of China ( 51835007, 51705360, 51775380, 51721003).

## Disclosures

The authors declare no conflicts of interest.

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