1. Introduction

High-accuracy contactless distance measurements are essential in various applications such as large-scale industrial manufacturing, alignment of particle accelerators, surveying, and environmental monitoring. Commercially available laser-based distance meters can provide instrumental accuracy in the sub-mm to mm level over distances ranging from a few meters to several km. However, the measurement accuracy in practical conditions is limited by the spatial and temporal variability of the refractive index ($n$) of air resulting from variations in the density and composition of the air.

Conventionally, the refractive index of air is modeled as a function of temperature ($T$), pressure ($P$), relative humidity ($RH$), and $\mathrm {CO}_2$ content ($x_c$) using empirical equations [1,2]. The standard approach to account for refractivity in electro-optical distance measurement (EDM) is to record the meteorological parameters with sufficient temporal resolution at a few locations along the measurement path—typically only near the instrument and, if higher accuracy is needed, also near the end point of the distance—to calculate so-called meteorological corrections using a forward modeling approach [3]. The relative humidity and even more so the $\mathrm {CO}_2$ content can often be neglected as they have much less impact than $T$ and $P$ for visible light and near-infrared. However, to properly account for refraction, the integral refractive index along the entire optical path is needed. Determining it from the few available meteorological measurements is very challenging. This is mostly due to the local differences and temporal changes in the spatial distribution of the temperature, and inherent uncertainties in the meteorological observations [3].

An alternative approach for refractivity compensation relies on atmospheric dispersion and uses simultaneous distance measurements at two or more wavelengths [4]. The method allows estimating the integral effect of the refractive index along the optical path and is robust to spatial and temporal fluctuations. Refractivity compensation using two wavelengths is typically known as the ‘two-color method’ [5]. It yields the estimated geometrical distance $D$ as a linear combination of the measured optical path lengths $\ell _{\lambda _1}$ and $\ell _{\lambda _2}$ at two wavelengths $\lambda _1$ and $\lambda _2$:

The coefficient $A$ is formally

The main challenges of this method are that (i) the measurement uncertainties of $(\ell _{\lambda _2} - \ell _{\lambda _1})$ are scaled by the typically large value of $A$, and (ii) the air is rarely completely dry and thus an error of $\Delta A \cdot (\ell _{\lambda _2} - \ell _{\lambda _1})$ is introduced in the compensation because the value of $A$ calculated only from the wavelengths differs from the correct value according to Eq. (2).

The first challenge can be mitigated by selecting a wavelength pair that ensures a comparably low value of $A$. This requires the wavelengths to be far apart, ideally one of them as close to the short wavelength end of the visible spectrum as possible and the other one in the near-infrared. While it is theoretically possible to achieve $A<10$ with such a choice, values of about 20 or higher result if both wavelengths are between 550 and 1600 nm that is practically useful for EDM. So, additionally very high accuracy (ideally µm level) for the optical path length measurements on the individual wavelengths, or at least for their difference, is needed in order to achieve high accuracy (mm- or sub-mm level) of the refraction compensated distance.

The impact of $RH$ on $D$ in Eq. (1) is approximately 3 times higher than on the individual observations $\ell _{\lambda _i}$. Neglecting $RH$ introduces a relative error of around $2\,$ppm on $D$ for $RH=100{\% }$ at temperatures below about 20$^\circ$C, and more at higher temperatures. Nevertheless, a relative uncertainty better than $0.1\,$ppm on $D$ can be achieved if $RH$ is monitored with a measurement uncertainty below $4{\% }$ [7], although this may be difficult in practical conditions and undermines the original idea of the two-color method, namely to compensate without requiring meteorological observations. Theoretically, the impact of $RH$ can also be eliminated by using optical path length measurements on more than two wavelengths. A three-color method has been proposed in the literature [8]. However, the measurement uncertainties in this approach are scaled by parameters on the order of $10^3$ in the visible and near-infrared region, making the three-color method so far unsuitable for practical applications. For many applications, challenge (ii) thus remains but affects the two-color method mostly by a moderate or even negligible degradation of the accuracy as compared to what can be achieved in dry air.

Recent technological advances in optical frequency combs (FC) have enabled highly precise and accurate measurements over long distances due to the intrinsically high stability of the optical frequencies and coherence lengths of several km [9]. Distance measurement using FC was first demonstrated in [10] by monitoring the phase-delay accumulated on the intermode beat notes obtained through direct photodetection of the FC. This pioneering work also demonstrated for the first time two-color distance measurements using a frequency-doubled FC source. Since then various alternative approaches using FC have been demonstrated, such as those based on interferometry [11], dual-combs [12,13], and time-of-flight [14] ranging.

Several other investigations on two-color solutions have also been presented in the literature [15–18]. In controlled conditions, two-color refractivity compensated distance measurements have achieved accuracies even better than the accuracy of the empirical refractive index equations and thus better than the accuracy achievable using meteorological observations and forward modeling, see [19,20]. Recently, a sub-mm two-color EDM using an intensity-modulated continuous wave approach has also been demonstrated over 5 km [21]. Typically, a frequency-doubled source or a combination of lasers centered at different wavelengths has been used in such investigations. However, the approaches lack spectral flexibility in selecting the wavelength pair and require very high accuracy and stability of the spatial alignment of the multiple beams [22]. A multi-color method ($>3$ wavelengths) has recently been demonstrated to mitigate the impact of varying $RH$ [23]. The authors used a SC source ranging from 1100 to 1700 nm and a dispersive interferometry-based approach to achieve high-precision distance measurements. The sweep speed of the optical spectrum analyzer (OSA) limits the data acquisition rate in this approach. The method requires high signal-to-noise (SNR) and fine resolution of the OSA for enhanced distance resolution [24].

We have previously demonstrated a new approach using intermode beats obtained from a SC for multi-wavelength distance measurement on cooperative targets [25,26] and hyperspectral LiDAR on natural surfaces [27]. This approach allows accurate and absolute distance measurements using the FC as a modulator [28]. The use of a SC source allows flexible selection of the wavelength pair and seamless extension to more than two wavelengths for multi-color distance measurements, and thus we chose to work with a SC herein rather than, e.g., use FCs at two different wavelengths. The SC used in our experiments covers the spectral range of 570-970 nm, taking advantage of the steeper gradient of the dispersion curve to achieve a low A factor. Additional co-alignment of multiple beams (at different wavelengths) is not necessary with our approach as the wavelengths are optically filtered from the same SC beam.

In this paper, we show an improved experimental design achieving a measurement precision on the order of $10^{-7}$ over a distance of 50 m. Distances are simultaneously measured on two spectral bands filtered from the SC spectrum. We further assess the observed distance precision as a function of the measurement distance and data integration time. Our results demonstrate long-term measurement stability over several hours using an internal compensation path. We compared our results with those obtained from a reference interferometer to estimate the distance accuracy of our measurements. For the first time, we also show two-color inline refractivity correction using the intermode beats obtained through the wavelength pair filtered from the SC.

The paper is organized as follows: the measurement principle and the experimental setup are described in Sec. 2. This includes a presentation of the lab setup for creating a significant but controlled refraction effect already at short distances. The experimental results on range precision, relative accuracy, long-term stability, and refractivity correction are presented in Sec. 3. In Sec. 4 we summarize the primary contributions with an outlook on the practical challenges and future directions of this work.

2. Methods

2.1 Experimental approach

We use a SC spectrally broadened from a 780 nm mode-locked fs laser (Menlo Systems C-fiber 780 SYNC) by transmitting through a photonic crystal fiber (Menlo Systems SCG1500). The generated SC spectrum ranges from 570 to 970 nm with an integrated optical power of around 26 mW. We set the pulse repetition rate of the mode-locked laser to $f_r\!=\!100\,\mathrm {MHz}$, locked to a rubidium frequency standard (SRS FS725) for accurate reference. Figure 1(a) shows an instantaneous spectrum of the resulting SC as observed by a commercial spectrometer (Thorlabs CCS 175).

 

Fig. 1. (a) Normalized instantaneous spectrum of the SC output; band-pass filtered spectrum of the (b) 590 nm and (c) 890 nm spectral bands; a schematic representation of the measurement principle showing the electrical power spectral density (PSD) of the intermode beat notes and phase shifts allowing distance estimation.

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As mentioned earlier, a low $A$-factor is desirable for the two-color method. Given the dispersion characteristics of air, this implies that the wavelength pair should be taken from the extreme ends of the SC spectrum. However, taking the power distribution of the available SC into account, we have selected 10 nm wide spectral bands centered at $\lambda _1\!=\!590$ and $\lambda _2\!=\!890\,\mathrm {nm}$ for further use herein. This resulted in a still acceptably low $A$ factor (approximately 36) while ensuring an adequate SNR. The A factor is calculated from the group refractive indices at $\lambda _1$ and $\lambda _2$ from the corrected Ciddor’s equation provided in [29]. The spectral bands are filtered using optical bandpass filters (BPF) centered at $\lambda _{1}$ and $\lambda _2$, see Fig. 1.

Corresponding electrical beat notes ($mf_r$, where $m=1,2,3,\ldots$) equally separated by $f_r$ are generated upon the photodetection of these spectral bands. In a detector noise-limited system, the measurement precision is expected to be directly proportional to the SNR of the electrical beat notes, which in turn is directly proportional to the received optical power [27]. The maximum beat note frequency is limited by the bandwidth of the photodetector (PD). Any of these beat notes can be used for distance measurements, where the higher frequency intermode beats offer better measurement precision due to their higher distance-to-phase sensitivity. Herein, we use an avalanche photodiode (APD) with a $-3\,\mathrm {dB}$ bandwidth of 1 GHz, and process the $10^\mathrm {th}$ harmonic mode, $f_M\!=\! 1\,\mathrm {GHz}$, to achieve high-precision distance measurements. For each of the wavelengths $\lambda$, the optical path length ($\ell _\lambda$) is calculated from the differential phase observations on $f_M$ between a fixed local reference path ($\phi ^\mathrm {ref}$) and a probe path ($\phi ^\mathrm {probe}$), see Fig. 1. The estimated distance $L_\lambda$ at each wavelength is derived from these phase observations as

2.2 Experimental setup

A schematic diagram of the experimental setup is shown in Fig. 2. The system depicted here comprises the optical and electronic parts along with a test track (comparator bench) on which a corner cube reflector can be moved to distances up to about 50 m, and refractivity variations can be introduced in a controlled manner.

 

Fig. 2. Schematic representation of the experimental setup (PCF: photonic crystal fiber; BE: beam expander; BS: beam splitter; PBS: polarization beam splitter; DM: dichroic mirror; BPF: band-pass filter; FL: focusing lens; NDF: variable neutral density filter; APD: avalanche photodiode; SH: mechanical shutter; M: plane mirror; HWP: half-wave plate; QWP: quarter-wave plate; CC: corner cube; CLK: clock; LO: local oscillator; A/D: analog-to-digital; I/Q: in-phase/quadrature.

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2.2.1 Distance meter

The fiber-collimated SC output beam has a diameter of about 4.5 mm. It is expanded in two stages to a final diameter of about 45 mm to maintain collimation over the entire probe path, and to reduce the impact of pointing fluctuations on the measurements. Taking potential chromatic aberration into account, we have chosen a reflective-type beam expander (BE, 2X magnification) and a set of achromatic doublets (5X BE) for this. The s-polarized light reflected from a broadband polarization beam splitter ($\mathrm {PBS_1}$, Thorlabs CCM1-PBS252) between the first and second stage of the beam expansion forms the local reference path, and the transmitted p-polarized light constitutes the probe path.

The reference path beam is spectrally split using a dichroic mirror (DM, Thorlabs DMSP650). The spectral bands, mentioned above, are filtered using the respective optical BPF, and each of them is fed to one APD. $\mathrm {R_1}$ and $\mathrm {R_2}$ represent the two RF signals (containing the electrical beat notes) for $\lambda _1$ and $\lambda _2$, respectively. On the probe path, the p-polarized light passes through a broadband quarter-wave plate (QWP) and is launched onto the comparator bench. An achromatic hollow corner cube reflector (CC, Newport U-BBR2.5-5S), mounted on a computer-controlled trolley that can move along the entire length of the comparator bench, reflects the beam along the same optical path. The beam becomes s-polarized after passing again through the QWP. It gets reflected, spectrally separated and directed onto the two probe APDs (one for each wavelength) in a configuration similar to the reference path. The RF signals from the probe APDs are denoted as $\mathrm {P_1}$ and $\mathrm {P_2}$.

An additional compensation path is included to monitor the phase errors introduced by small drifts in the electronic setup, including variations of the propagation speed within the high-frequency cables. This approach is similar to the experimental configuration described in [28] and used there to achieve stable measurements over several hours. We use a 70:30 beam splitter (BS) to redirect $30{\% }$ of the SC power onto the compensation path. This beam passes through a broadband half-wave plate (HWP) and finally the same elements as the probe beam before reaching the probe APDs. The optical power illuminated on each APD is controlled using variable neutral density filters (NDF) and is adjusted to around 12 µW to guarantee operation within the linear region of the APDs. Two mechanical shutters $\mathrm {SH_1}$ and $\mathrm {SH_2}$ are used to alternate between the probe and compensation path measurements.

The 1 GHz beat note is filtered from the respective RF signals using analog BPFs centered at $f_M$. The filtered beat notes are downconverted to an intermediate frequency of $f_\mathrm {IF} = 400\,\mathrm {kHz}$ upon mixing with a signal generated from a local oscillator (LO). The downconverted signals are again filtered using analog BPFs, now centered at $f_\mathrm {IF}$, to reduce SNR degradation by noise aliasing. The signals are subsequently digitized simultaneously using a 4-channel analog-to-digital converter (ADC, Spectrum Instrumentation M4i.2234-x8). A digital in-phase/quadrature (I/Q) phasemeter is implemented to retrieve the phase observations on each channel. Using Eq. (3), the optical path length at each wavelength is then estimated from the observed phase differences. Accurate clock references to the LO and ADC are also obtained from the Rb frequency standard.

To later also assess the accuracy of the measured distances, a He-Ne Doppler interferometer (Agilent 5519A) is included in the setup. It provides reference values of the trolley’s displacement. The interferometer uses a different retroreflector mounted below the CC on the same automated trolley. Although Abbe’s principle [32] is therefore not maintained, the spatial separation of approximately 5 cm between these two reflectors is small enough such that potential (minute) tilt or orientation changes of the trolley are negligible, see Figure 3(a). The probe and interferometer beams are both aligned to be sufficiently parallel along the comparator bench such that also the potential cosine error [33] is negligible.

 

Fig. 3. Picture of the (a) computer-controlled trolley on a linear 50 m long comparator bench fitted with the retroreflectors for SC-based distance measurements and the reference He-Ne interferometer, (b) refraction modifying element i.e., 9 m long sealed cylindrical tube placed on the comparator bench.

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2.2.2 Refraction modifying element

Air conditioning keeps the temperature in our lab constant within about $0.5\,^\circ \mathrm {C}$ and relative humidity within $3{\% }$, or better. Spatial variations of the refractive index of the air in the lab are small (on the order of 1 ppm or less), and temporal variations are mostly caused by natural changes of the barometric pressure outside the building. In order to produce refractive index variations of experimentally useful magnitude along the probe path in a controlled manner, we have included a refraction modifying element on the comparator bench for the experiments described in Sec. 3.4. Taking into account the dimensions of the lab, the previously found precision of the optical path length measurements (a few micrometers), the $A$ factor, and the desired flexibility to easily add or remove the element, we have decided for a design which can create a relatively large change of the refractive index (on the order of 45 ppm or more, as compared to the ambient conditions in the rest of the lab) over a short distance.

We initially considered introducing refractivity changes through temperature variations along the probe path. However, it required a temperature difference of about 45 $^\circ \mathrm {C}$ or more to create the refractivity change mentioned earlier. Instead, the solution that we found practical and have finally used in our experiments was based on creating a local change of the barometric pressure within a sealed cylindrical tube with clear optical windows on both ends. We used a 9 m long sealed steel tube connected to an electrical air pump that depressurizes the tube by extracting air from the inside. We obtained the experimental results shown in Sec. 3.4 by carrying out measurements while this tube was temporarily placed on the comparator bench between the distance meter and the trolley with the CC. A picture of the computer-controlled trolley and the sealed cylindrical tube is shown in Fig. 3.

Our experiments were conducted under stable lab conditions at 20$^\circ$C, 50% relative humidity, 410 ppm $\mathrm {CO_2}$ content, and an average barometric pressure of around 960 hPa. The meteorological changes over the maximum duration of our experiments (i.e., around 5 hours) were $\Delta T \leq 0.3^\circ \mathrm {C}$, $\Delta P\leq 2\,\mathrm {hPa}$, $\Delta RH \leq 1.5{\% }$ and $\Delta \mathrm {x_c} \leq 30\,\mathrm {ppm}$, which resulted in meteorologically-induced distance errors (driven by $\Delta T$ and $\Delta P$) that do not exceed 0.4 ppm. Distance errors originating from potential physical deformations of the comparator bench during the experiments are negligible compared to the target accuracy of our experiments. So, noticeable distance changes were only introduced in a controlled manner by moving the trolley and by using the depressurized tube.

3. Results

3.1 Measurement precision

In this section, we present experimental results on the estimated measurement precision as a function of distance and data integration time. The CC was moved from a distance of 10 m to 50  m in equidistant steps of 5 m and kept stationary during the data acquisition at each distance. The variable NDFs were adjusted at each step to maintain approximately the same intensity of the illumination on all APDs and for all distances of the CC. This helped to reduce potential errors related to the relative power-to-phase coupling [28].

Figure 4(a, b) shows the time series of distance deviations ($\Delta L_\lambda$) from the initial measurement simultaneously acquired on $\lambda _{1}, \lambda _2$ for the CC at a (one-way) distance ($L$) of 10 and 50 m, respectively, from the instrument. Each of these data points was obtained by averaging over an integration length of $\tau _w$. Unless otherwise indicated, we use $\tau _w\!=\!27\,\mathrm {ms}$ to compute our results and denote this particular integration time as $\tau _w'$ henceforth. We quantify the measurement precision using the empirical standard deviation $(\sigma _{\Delta \mathrm {L}})$ calculated from the 200 data points of the respective time series. The standard deviation is around 3 µm at 10 m and 9 µm at 50 m on both wavelengths. The empirical standard deviation of $(\Delta \ell _{\lambda _2} - \Delta \ell _{\lambda _1})$ is around 4 µm at 10 m and 8 µm at 50 m. The data were processed in near real-time, where each measurement epoch takes about 1.3 s of computation time on a standard PC to process the data acquired on both wavelengths. The processing speed could be significantly improved by more powerful or dedicated hardware and parallel processing if a higher acquisition rate were required.

 

Fig. 4. Time series of the observed distance deviations at (a) 10 m and (b) 50 m where each measurement point is averaged over 27 ms, (c) empirically estimated measurement precision as a function of the distance ($L$) to the corner cube. The estimated fit to the observations from Eq. (4) is shown as dotted line.

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Figure 4(c) illustrates the empirically estimated standard deviation $\sigma _{\Delta \mathrm {L}}$ for each color and distance. To support the interpretation, we have calculated the $95{\% }$ confidence intervals of these empirical standard deviations and plotted them as error bars in the figure. Assuming that the standard deviation $\sigma _{\Delta \mathrm {L}}$ can be modeled as a function of $L$ according to the approach typically used for EDM [34], we fitted tentatively a model as follows to the data plotted in Fig. 4(c):

Here, $\sigma _0$ represents the measurement noise floor including error sources from mechanical vibrations and other optical and electronic instabilities, and $k_L$ represents the coefficient of the distance-dependent contributions, which include, among others, impacts of beam divergence, atmospheric attenuation, and beam pointing. From our data, we estimated these values to be $\sigma _0 \approx 2$ µm and $k_L \approx 0.18$ µm/m. The fitted model is also depicted in Fig. 4(c) as a dotted line. The figure shows that this model is a good approximation to the data, but there are statistically significant deviations; for 3 rather than the theoretically expected 1 of the 18 data points, the fitted model is not within the $95{\% }$ confidence interval of the empirical standard deviations. Although this indicates that the model does not capture all dependencies, e.g., it does not account for the differences in power at the different wavelengths or minor potential variations of the meteorological conditions during the acquisition of 200 measurements per data point. We consider the model adequate for the present investigation and leave a more thorough analysis to the future (see also below). The magnitude of $k_L$ suggests the possibility of achieving sub-mm level precision over several km in controlled conditions. However $k_L$ may increase under practical outdoor conditions, considering more significant air turbulence and scattering due to atmospheric particles. This needs to be further investigated once outdoor measurements are possible with a successor of our current setup.

We also investigated the dependence of $\sigma _{\Delta L}$ on the data integration time, as shown in Fig. 5(a,b) for 10 and 50 m respectively. The underlying raw data are a time series of distance observations acquired with a data rate of approximately 78 MS/s during around 56 s (limited by the ADC buffer memory). These data were then processed offline using non-overlapping moving average windows of length $\tau _w$. For each chosen value from $\tau _w\!=\!0.04\,\mathrm {ms}$ to $\tau _w\!=\!1\,\mathrm {s}$ we calculated the empirical standard deviation $\sigma _{\Delta L}$. The log-log plot of the results for the 10 m distance, see Fig. 5(a), indicates a slope of around $-1/2$ up to 10 ms, implying that the measurement processes are dominated by white noise [35] at this distance. The slower improvement of precision with integration time beyond 10 ms suggests the presence of slow drifts at sub-Hz to Hz levels. They may originate from mechanical vibrations of the experimental setup and temporal variations of the optical power on the individual spectral channels.

 

Fig. 5. Empirically estimated distance precision as a function of data integration time $\tau _w$ at (a) 10 m and (b) 50 m.

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At a distance of 50 m, see Fig. 5(b), white noise characteristics are observed up to integration times of $\tau _w$ = 1 ms. The improvement of precision with increasing integration time then reduces, and the precision hardly changes for integration times from 40 ms to 0.1 s, suggesting the presence of correlated noise in the data [35]. This effect may be attributed to temporal beam-pointing fluctuations at these timescales. For integration times beyond 0.1 s, the measurement precision again shows white noise-dominated characteristics and improves further with longer integration times. The range precision can be improved to around 5 µm at 50 m using $\tau _w = 1\,\mathrm {s}$. The results suggest the possibility of achieving a measurement precision below $0.1\,\mathrm {ppm}$ using integration times of about 0.5 to $1\,\mathrm {s}$.

3.2 Relative accuracy

To evaluate the measured distance accuracy, we compared our results with those obtained using the reference interferometer. We moved the CC back and forth in 3 cycles between 49 and 50 m at equidistant steps of 1 cm. At each step we averaged a set of 5 data points, where each data point was obtained in real-time by integrating over $\tau _w'$.

The residuals obtained by subtracting our measurements from the reference measurements are shown in Fig. 6(a). They are all within $\pm 45$ µm and dominated by systematic cyclic errors which are induced by optical or electrical cross-talk. Due to their systematic nature they can be mitigated by calibration [27,36]. The observed cyclic error period is around 15 cm, which corresponds to the range ambiguity $\Lambda _M$, see sec. 2, of the 1 GHz beat note used in our experiments. We determined a correction by fitting a sinusoidal function with a fixed period of $\Lambda _M$ to the residuals from Fig. 6(a). The post-calibration residuals obtained by subtracting the correction function are shown in Fig. 6(b). The root-mean-square error (RMSE) of these residuals is around 8 µm on both wavelengths, corresponding to a relative accuracy on the order of 0.1 ppm at a distance of 50 m. These results indicate that the cyclic errors can be corrected down to the precision-limited performance. Since the processes that drive these cyclic errors depend on the distance and signal power, the amplitude and phase of the derived calibration function are also dependent on the range. A practically applicable calibration would therefore follow the established process of calibrating cyclic deviations of EDM instruments, see e.g. [34]. It would require reference measurements at closely spaced distances across the entire measurement range and additionally have to include the detected probe power as an input along the estimated range if the power varies over time.

 

Fig. 6. (a) Residuals between the measured ($L_\lambda$) and the reference interferometer ($L_\mathrm {HeNe}$) showing cyclic error and the sinusoidal calibration fit between 49-50 m, (b) residuals after calibration.

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3.3 Long-term stability

As described in Sec. 2.2, we have implemented an internal compensation path to ensure the stability of our measurements over extended periods. Regulated by the two mechanical shutters $\mathrm {SH_1}$ and $\mathrm {SH_2}$, the distance observations are acquired on the probe ($L^\mathrm {probe}$) and compensation ($L^\mathrm {comp}$) path alternatively. $L^\mathrm {probe}$ is estimated using Eq. (3) from the simultaneous phase observations on the reference and probe path. Equivalently, $L^\mathrm {comp}$ is estimated from the reference and internal compensation path. A low-pass filtered (moving-averaged) output $\bar {L}^\mathrm {comp}$ is used to rectify the slow bias errors on $L^\mathrm {probe}$.

Figure 7(a) shows the deviations in the probe and compensation path measurements monitored over 5 hours. The CC is placed at a close distance (about 2 m) to reduce the distance-dependent errors and assess the performance of the internal compensation. Measurements are taken approximately every 4 s, each averaged over $\tau _w'$. The observations of $L^\mathrm {comp}$ are shown in a lighter color tone with an offset of $-25$ µm for better clarity. The 15-point moving-averages $\bar {L}^\mathrm {comp}$ are shown as black dashed lines. The moving-average window size corresponds to a time constant of around 59 s. Deviations of around 60 and 40 µm can be observed on the $L^\mathrm {probe}$ measurements at $\lambda _1$ and $\lambda _2$ respectively, depicting significant temporal variations over the measurement duration. The variations of the compensation measurements $L^\mathrm {comp}$ exhibit a similar trend.

 

Fig. 7. Distance deviations ($L\approx 2\,\mathrm {m}$) monitored over 5 hours showing (a) long-term drifts in the probe path measurements $L^\mathrm {probe}$ and corresponding measurements from the internal compensation path $L^\mathrm {comp}$ (shifted by $-25$ µm for clarity), (b) compensated measurements ($L^\mathrm {probe} - \bar {L}^\mathrm {comp}$).

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The distance observations representing long-term stability are then calculated as ($L^\mathrm {probe} - \bar {L}^\mathrm {comp}$) and are shown in Fig. 7(b). The significant temporal drifts are eliminated, and the compensated measurements are almost stable over time. The estimated standard deviation and mean offset of the resulting time series is around 4 µm. The empirical standard deviation of $(\ell _{\lambda _2} - \ell _{\lambda _1})$ is around 5 µm. The minor residual drifts are correlated with the small changes in the barometric pressure ($\approx 2\,\mathrm {hPa}$) and temperature ($\approx 0.3\,^\circ \mathrm {C}$) during the measurement. Since the remaining residuals after internal compensation are already at the level of the underlying measurement noise floor, we decided not to include additional model-based meteorological correction for simplicity. Nevertheless, these variations will be compensated by the subsequently described refractivity compensation without using meteorological observations.

3.4 Refractivity compensation

We conducted further experiments to demonstrate inline refractivity compensation using intermode beating. The 9 m long sealed tube was placed on the comparator bench to create pressure-induced refractivity changes along the probe path. The CC was placed at a distance of 15 m (i.e., $L=15\,\mathrm {m}$), and the tube was positioned between the distance meter setup and the CC, leaving a gap of 3 m on each side. Figure 8(a) shows a time series of the air pressure ($P$) variation attained inside the tube. The pressure starts decreasing when the connected air pump is switched on and gradually increases again when the pump is switched off (driven by small leaks in the pipe assembly). The maximum $\Delta P$ achieved is approximately $-166\,\mathrm {hPa}$, and the temperature change ($\Delta T$) within the tube is about 0.2 $^\circ \mathrm {C}$ following a similar pattern as the pressure change. The resulting refractive index change $\Delta n$ is around $-45\times 10^{-6}$ inside the sealed tube, which is primarily dominated by the pressure variations.

 

Fig. 8. Time series showing (a) pressure variations of $P$ monitored within the tube, (b) corresponding distance deviations measured using the reference interferometer $\Delta L_\mathrm {HeNe}$, (c) simultaneously acquired optical path length deviations on the two wavelengths $\Delta \ell _{\lambda _1}$ and $\Delta \ell _{\lambda _2}$ (which is shifted by $-0.1\,\mathrm {mm}$ for clarity), two-color refractivity compensated measurements $\Delta D$, and its 21-point moving-averaged result $\overline {\Delta D}$.

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Figure 8(b) shows the apparent distance changes $\Delta L_\mathrm {HeNe}$ of up to $-0.4\,\mathrm {mm}$, observed using the reference interferometer. This is consistent with our theoretical predictions from the empirical equations [1,2] and can be estimated as $\Delta n$ times the length of the refraction modifying tube, i.e approximately $-45 \cdot 10^{-6} \times 9\,\mathrm {m}$. As expected, $\Delta L_\mathrm {HeNe}$ follows the same pattern as the pressure variations introduced within the pipe.

The observed optical path length changes ($\Delta \ell _{\lambda _1}, \Delta \ell _{\lambda _2}$) with respect to the initial value at each wavelength are shown in Fig. 8(c), where $\Delta \ell _{\lambda _2}$ is shifted by $-0.1\,\mathrm {mm}$ for clarity. The experiment implements differential measurements to the internal compensation path to ensure long-term stability. The variations $\Delta \ell _\lambda$ of the measured path lengths also exhibit a similar pressure-driven pattern and distance deviations as observed for the reference interferometer in Fig. 8(b). The estimated change in geometrical distance $\Delta D$ derived from Eq. (1) using the observations $\ell _\lambda$ is depicted in Fig. 8(c). The resulting estimated distance shows that the induced drift apparent in $\Delta \ell _\lambda$ and in the reference interferometer measurements is suppressed, thereby demonstrating an effective two-color refractivity compensation. As anticipated, the measurement uncertainty is scaled by the $A$ factor. The empirical standard deviation $\sigma (\Delta D)$ before filtering is around 160 µm. The noticeable sharp peak in $\Delta D$ at around 7 minutes is likely caused by a large beam pointing related phase error when the pump is switched on. The $\Delta D$ observations are averaged using a 21-point (about 60 s) moving window to eliminate the higher frequency fluctuations. The filtered data ($\overline {\Delta D}$) is also shown in Fig. 8(c). The empirical standard deviation $\sigma (\overline {\Delta D})$ is around 85 µm, indicating a relative distance precision on the ppm-level without using any external meteorological observation. Each data point is based on only $27\,\mathrm {ms}$ integration time followed by a break of almost 3 s, introduced in this experiment because of data storage and raw data processing limitations of our current experimental setup. Based on the noise analysis presented earlier (see. Sec. 3.1), we expect that the same standard deviation would be achieved by averaging 21 measurements collected directly after each other, i.e. by averaging over less than 0.6 s, and a further improvement can be achieved by averaging longer.

Due to non-dry conditions the A-value calculated from the wavelengths (see Sec. 1) is wrong by about $\lvert \Delta A \rvert \approx 0.031$ for the maximum $\Delta P$ observed herein. The dispersion measurement ($\ell _{\lambda _2} - \ell _{\lambda _1})$ is around 10 µm at the corresponding time. Consequently, the impact of humidity on the distance measurements is on the order of $\Delta A \cdot (\ell _{\lambda _2} - \ell _{\lambda _1}) \approx$ 0.3 µm, and can thus be neglected in comparison to our measurement precision. The residual errors in $\overline {\Delta D}$ over the expected noise-related uncertainty scaled up by the A factor may originate from wavelength-specific beam pointing and optical power instability [37]. A comparative power spectral analysis of these residuals and detected signal amplitude shows a similar higher density in the mHz range. This indicates that the remaining low-frequency variations in $\overline {\Delta D}$ at time scales of minutes are likely induced by pointing and optical power variations. Nevertheless, using our current experimental setup, we were able to achieve refractivity-corrected distance measurements on the ppm level over the assessed range.

While this relative precision indicates promising perspectives towards the extension of the method to longer practically useful ranges, some additional challenges should still be considered. On the one hand, the distance errors induced by refractivity variations become more noticeable when the measurement range extends beyond a few hundred meters. Considering e.g. a range of 500 m, i.e. more than 30 times longer than our current experiments, a refractivity change or error of 10 ppm that can be expected in practical conditions would lead to a single-wavelength distance error of about 5 mm. Considering the distance-dependent coefficient $k_L$ for the precision on individual wavelengths as modeled through our experiments, the two-color compensated precision would be better than 5 mm using integration times of tens of ms and can be further improved by longer averaging, thus still providing useful compensation. However, increased turbulence over longer distances may lead to further degradation of performance led by pointing and power fluctuations. Extending the averaging time window can partly reduce these errors, but more specific mitigation procedures should also be considered and further investigated.

Various techniques have been demonstrated in the literature to experimentally analyze [28], suppress [38], and characterize the pointing and power-to-phase conversion effects [39,40]. An amplitude fluctuation-based data selection has been shown to be effective in eliminating power-to-phase errors for measurements over several km, which are strongly affected by atmospheric turbulence [41]. The recent technological advances in developing ultra-low-noise SC sources (characterized by low relative intensity noise and high phase coherence) also contribute towards reducing the phase errors caused by the inherent power instabilities [42]. Further exploring the potential solutions to mitigate these effects is essential to develop the proposed approach towards working implementations in real conditions. While we have investigated the relevant sources of uncertainties to assess the fundamental performance of our approach, we have not attempted a comprehensive uncertainty quantification according to the ‘Guide to the expression of uncertainty in measurements’ (GUM) [43]. We leave this for the time when we have moved from a lab prototype of our system to a more mobile measurement platform.

4. Conclusion and outlook

We have investigated the potential of coherently broadened fs-lasers for refractivity-compensated distance measurement. The developed experimental platform achieves highly precise and accurate distance measurements at multiple spectrally filtered bands from a coherent supercontinuum. Distances are estimated from the differential phase delay accumulated on the 1 GHz beat note. For single-wavelength measurements integrated over 27 ms, we obtained standard deviations below 10 µm over a distance of 50 m, and found that this standard deviation contains a length-dependent contribution of below 0.2 ppm. Additionally, we observed that the precision of the measurements can be improved by integrating over longer times; the standard deviation generally decreases with integration time $\tau _w$, and is even inversely proportional to the square root of the integration time for certain ranges of $\tau _w$. Compared to a reference interferometer, our measurements show a relative accuracy on the order of 0.1 ppm after calibrating the systematic cyclic deviations. Additional observations from an internal compensation path are used to achieve measurement stability over several hours. We further demonstrated inline refractivity compensation using the two-color method. Pressure-induced refractivity variations were created along the measurement path, and the refractivity corrected distance measurements show an empirical standard deviation less than 0.1 mm at 15 m when the optical path length change is about 0.4 mm. Phase errors originating from power and pointing instability are the current limitations of our experimental setup. Further investigations are necessary to calibrate or suppress the power-to-phase conversion characteristics. The experimental setup can be easily extended for simultaneous multi-color distance measurements by including measurements from additional photodetectors. This may enable a promising technological basis for practical long-distance measurements. It may also be possible to apply this approach even for measurements to non-cooperative targets if the wavelength dependent scattering and surface penetration can be distinguished from atmospheric dispersion. Investigating this is left for future work.