In this work, we demonstrate two-color inline refractivity compensation in a heterodyne synthetic wavelength interferometer for a measurement of absolute distances over several hundred meters with sub-millimeter accuracy. Two frequency-doubled Nd:YAG lasers with a coherence length of more than 1 km are used as light sources. Direct SI traceability is achieved by controlling the lasers' frequency difference in the radio frequency regime. The resulting synthetic wavelengths at 532 nm and 1064 nm are used for the absolute distance measurement and dispersion-based inline refractive index compensation. A standard deviation of 50 *μ*m is achieved for distances up to 864 m. This performance corresponds to a standard deviation of the observable, the difference of the four optical wavelengths, on a sub-nanometer level. Comparison against white light interferometry confirms sub-millimeter accuracy over this distance. Temporally resolved data over 864 m provide quantitative insights into the influence of chromatic beam paths.

Bridges, sites of nuclear power plants, and mountainous areas in which landslides may occur are all areas in which safety is of critical importance. These areas are monitored to assess the changes of a few millimetres or smaller per year which take place over distances of up to several hundred meters. Driven by modern, high-frequency electronics and data acquisition systems, radio-frequency (RF) modulation based techniques^{1} and frequency-sweeping interferometry^{2,3} have evolved considerably. Frequency combs enable the use of novel measurement strategies based, for example, on spectral interferometry,^{4–6} on dual-comb interferometry,^{7–9} and on cross-correlation time-of-flight.^{10} However, to be highly accurate outdoors, a distance measurement has to overcome one fundamental challenge: In air, all optical measurements are sensitive to the optical path length (OPL) *d* ≡ *nl*, i.e., the geometrical path *l* multiplied by the refractive index of air *n*. The refractive index can be approximated from the measured air temperature *t*, pressure *p*, partial pressure of water vapor *p*_{w}, and CO_{2} content *x* using the empirical Edlén equation

*K*(*λ*), *D*(*t*, *p*, *x*), and *g*(*λ*) represent analytical expressions such as those derived by Bönsch and Potulski.^{11} However, given typical inhomogeneities, sufficiently accurate temperature and humidity measurement setups that use networks of sensors over long outdoor paths are difficult to realize and adapt.^{12} Using additional equipment, it is possible to determine the effective temperature and humidity in the beam path by means of spectroscopic methods^{13–16} and by means of the speed of sound.^{17} However, the method that can be best adapted is a derivation of the geometric path *l* from the OPLs *d*_{1,2} with two different well-known vacuum wavelengths *λ*_{1} and *λ*_{2} by^{18}

with the parameter

In the limit of dry air, i.e., for *p*_{w} → 0, the factor *A* depends only on the vacuum wavelengths but not on the temperature, pressure, or CO_{2} content of the air. Inserting the empirical Edlén equations with the structure in Eq. (1) in Eq. (3) leads to

This measurement principle can also be applied in humid air. Only the water vapor pressure *p*_{w} must be determined in addition. In this more general case, the geometric length can be derived by^{19}

from *p*_{w} and the observed OPLs *d*_{1} and *d*_{2}. Indoors, the impressive stability of refractivity-compensated distance measurements on the order of 10^{−8} has been demonstrated.^{20–22} The authors recently reported on a 3D-capable displacement interferometer that achieved a relative uncertainty of 1 × 10^{−7} (Ref. 23) in industrial environments. Early work on absolute refractivity-compensated measurements was based on modulation techniques using He–Ne and He–Cd lasers^{24} and on frequency-doubled femtosecond mode-locked laser sources.^{25}

A disadvantage to this method is the fact that any uncertainty in the difference (*d*_{2} − *d*_{1}) is scaled by the factor *A*. We apply heterodyne multi-wavelength interferometry in order to obtain a highly accurate determination of the OPLs *d*_{1,2}. The phase Φ of a conventional interferometer in air depends on the length *l*, the vacuum wavelength *λ*, and the refractive index of air *n* and is given by Φ = 4*πnl*/*λ*. The non-ambiguous measuring range of an interferometer is limited to *λ*/2. Using two or more different wavelengths *i*, *j* = 1, 2, this range can be extended by using “synthetic wavelengths,” which are defined by the phase differences

with the synthetic wavelength Λ_{ij} = *λ _{j}λ_{i}*/(

*λ*−

_{j}*λ*). For

_{i}*λ*→

_{i}*λ*, the effective refractive index

_{j}*n*=

_{ij}*n*− (

_{i}*n*−

_{j}*n*)

_{i}*λ*/(

_{i}*λ*−

_{j}*λ*) for the synthetic wavelength approaches the group refractive index. Any uncertainty scales with the factor Λ

_{i}_{ij}/

*λ*compared to a single wavelength measurement. The OPLs derived by synthetic wavelength measurements can be intrinsically refractivity-compensated using

^{26}

where

As a light source of high coherence, two frequency-doubled Nd:YAG lasers (Innolight Prometheus 20) were chosen that generate laser light in the infrared at 1064 nm and the frequency-doubled green at 532 nm. The frequency difference of the two infrared wavelengths of the two lasers is stabilized by means of a phase-locked loop (PLL) to *f*_{PLL} = 20.01 GHz, enforcing a frequency difference of 2*f*_{PLL} = 40.02 GHz in the green. This generates synthetic wavelengths Λ_{s,i} = *c*/(*i *×* f*_{PLL}) of ≈15 mm for the infrared (*i* = 1) and ≈7.5 mm for the green light (*i* = 2). In addition, the frequency of part of the infrared beams is shifted using an acousto-optic frequency shifter (AOM) by 190 MHz, thus creating an additional synthetic wavelength of ≈1.5 m. For this reason, a pre-value of the distances must be known better than 37.5 cm for a successful deconvolution. A heterodyne detection scheme is pursued that enables parallel detection of the various signals of one color and that reduces the background noise level. To this end, several AOMs are used to generate pairs of measurement (M) and local oscillator (LO) beams, giving heterodyne signals of 5 MHz and 7.5 MHz (both green and infrared) and 8 MHz for the additional infrared wavelength. The beam generation is described in depth in Ref. 26 and summarized in Fig. 1. In our system, the ratio Λ/*λ* amounts to ≈14 100 and the *A* factor to ≈ 21.2 for the group refractive index for the 532/1064 nm combination. Hence, due to the design of our system, uncertainties in the optical wavelength measurement are scaled in our system by design by a factor of ≈300 000.

The prepared beams are transferred to the separate interferometer head by means of four polarization-maintaining fibers. A diagram of the interferometer optics is given in Fig. 1(b). The 1064 and 532 nm M beams are superposed. The collinear beams are then divided by a non-polarizing beam splitter cube (BS1) and directed through a broadband polarizing beam splitter (PBS) and a Fresnel rhomb (FR) as a broadband quarter-wave plate. One beam is expanded to a beam diameter of 25 mm by a pair of achromatic lenses with focal lengths of 25 mm and 250 mm. This beam is used as the probe beam, while the other beam is reflected back by a mirror to form the reference path for the heterodyne interferometer. Thus, any phase change in the fibers or in the light source can be compensated. The returning beams with s-polarization are reflected in the PBS. The corresponding s-polarized LO beams interfere in BS3 and BS4 with the probe and reference beams. After the interference filters, two InGaAs photo diodes detect the beat notes in the infrared, while two silicon PIN diodes are used for the green signals. External transimpedance amplifiers (DHPCA-100, Femto) convert the photo currents to voltages. These are then digitized by a 16 bit 100 MSample/s analogue digital converter with integrated field programmable gate arrays (FPGA) (SIS 3302, Struck). The FPGAs process the raw data in real time and calculate the sine and cosine of the interferometer phases of the heterodyne carrier frequencies with an integration time of 10 *μ*s.^{27} For the outdoor measurements, 65 536 samples were measured every 4 s. To account for turbulence, the data were filtered based on an amplitude threshold. The remaining data were divided into blocks of 1024 samples, and the average phases and amplitudes of these blocks were derived and stored for offline processing.

Using Eq. (6), the SI definition of the meter *l *=* c*Δ*t*/*n* (with Δ*t* representing the propagation time) can be rewritten as

for the two-color observation with synthetic wavelengths. The geometric length *l* is deduced by Eq. (7). SI traceability is hence ensured by the Edlén model used, by the measurement of the synthetic phases Φ_{s,i} and of the water vapor pressure *p*_{w}, and by the traceable generation of *f*_{PLL} in the RF band. The uncertainty of the Edlén model itself is in the order of 1 × 10^{−8}.^{11} An uncertainty of 100 Pa in water pressure corresponds to a relative length uncertainty of 1 × 10^{−7}.^{19} Using a standard PLL chain and a commercial Rubidium frequency standard, the relative uncertainty of *f*_{PLL} can be kept below 1 × 10^{−9}. In practice, deviations of *f*_{PLL} are immediately recognized by excessive measurement deviations. The overall uncertainty of this measurement principle, however, is governed by the uncertainty scaling due to combination of the two-color method and the synthetic wavelength principle used, leading to an unfavorable scaling of any uncertainty of the optical phase measurement in the order of approx. 300 000. Our long-distance measurement realization thus provides a direct, primary, but also challenging link to the SI definition of the meter.

The performance of the long-distance interferometer was investigated at the Nummela standard baseline in Finland. The baseline has six pillars positioned at 0 m, 24 m, 72 m, 216 m, 432 m, and 864 m that are calibrated by means of the Väisälä white light interferometer^{28} with a length-independent combined uncertainty (coverage factor *k* = 2) of $Uk=2ref=0.26\u2009mm$. The interferometer was placed on an auxiliary pillar 2.93 m in front of the 0 m position. Several wired environmental sensors were distributed between 0 m and 81 m. A more extended real-time monitoring system was not available on site during the measurements. Since the Nummela baseline is almost fully covered by trees and the sky was clouded for most of the time, the temperature and humidity distribution could be considered relatively homogeneous for the campaign. The measured pressure was adjusted to the average height of the path. The distances to the five pillars between 24 m and 864 m were measured several times during a five-day period.

The repeatability of our system in outdoor measurements is characterized in Fig. 2. The relevant reference point is the center of the mounting tribrach. Figure 2(a) shows the variation of the offset during the outdoor measurement campaign related to the first measurement. The offset for the 532 nm wavelength has a standard deviation of 35 *μ*m. This magnitude is in good agreement with the expected reproducibility of the tribrach mounting.^{12} By contrast, the 1064 nm offset shows variations of more than 0.2 mm. The most probable explanation for this “chromatic” effect is an environment-dependent polarization cross talk. Some of the beam splitters were used for both infrared and green light simultaneously. Under field conditions, they seem to alter their polarization properties, in particular in the infrared. This induces a time-dependent polarization cross talk of the infrared M and LO beams and thus a changing offset. This interpretation was supported by laboratory verification experiments afterwards. To circumvent this problem, differential measurements were performed: the reflector was mounted on the 0 m pillar, then on the target pillar, and again on the 0 m pillar. The standard deviation *σ* of two measurements at 864 m is shown in Fig. 2(b). The experimental data were evaluated using both the refractive index from the environmental parameters and the two-color inline refractivity compensation according to Eq. (7). The PLL stabilization between both Nd:YAG lasers showed oscillations of 440 Hz with an amplitude corresponding to 10^{−10} in laser frequency. This limits the achievable standard deviation for integration times below 10 ms. For integration times above 1 s, the standard deviation *σ* of the Edlén-based results decreases only slightly, probably due to temperature variations that could not be resolved by the temperature sensors. The standard deviation of the two-color inline compensated lengths decreases to less than 50 *μ*m for 10 s of integration time or a relative uncertainty of 6 × 10^{−8}. Given the uncertainty scaling of a factor of ≈300 000, this value corresponds to a standard deviation of the difference of the four optical wavelengths on the order of 170 pm. This remarkable performance is one important prerequisite for high accuracy to be achieved in the absolute measurement.

To investigate the achievable accuracy, absolute distance observations can be compared to the white-light reference values available at Nummela. The deviations Δ*l*_{2-color} calculated according to Eq. (7) from the reference values are compiled in Fig. 3(a). The standard deviation is in the range from 0.3 mm at 24 m to 0.6 mm at 864 m. Due to the unfavorable uncertainty scaling, the observed standard deviation of 0.6 mm at 864 m corresponds to 2 nm only in the difference between the single wavelength results over all observations. The mean values are in reasonable agreement with the reference values. It should be noted that polarization crosstalk can be expected to lead to a length-dependent periodic error, a so-called “cyclic-error.” A cyclic error in the order of 0.2 *μ*m in one color would be scaled by the *A* factor to a deviation in the order of several centimeters. After applying the “short-term” offset subtraction as described in the previous paragraph, deviations in this magnitude were not observed. Nevertheless, a smaller cyclic error cannot be excluded based on the experimental results. It would contribute to the observed standard deviation over the complete dataset. The OPLs in the green and infrared can also be processed conventionally using the environmental data and the Edlén equation. In general, the measured values are in very good agreement with the reference values although all environmental sensors were only located till 100 m. The systematic outlier at 216 m could indicate a substantial eccentricity of the tribrach used there. The standard deviation $\sigma Edle\xb4n$ of the Edlén-based results for green and infrared light was calculated and plotted in Fig. 3(c). The length dependency of $\sigma Edle\xb4n$ can be approximated by $(35\u2009\mu m)2+(2.4\xd710\u22127\xd7l)2$. The constant contribution of 35 *μ*m is consistent with the short-term standard deviation of the offset. The scale parameter of 2.4 × 10^{−7} corresponds well to a temperature uncertainty of 0.24 K. This appears to be reasonable for the stable conditions during the measurements.

The use of sensors that have a limited temporal and spatial resolution can lead to false confidence in data quality and to wrong conclusions. Within a single observation measurement, the conventionally analyzed interferometer data in Fig. 4(a) suggest an apparent shift of the pillar position by 80 *μ*m between 0 and 5 min. Given the known extraordinary stability of the reference pillars and the good correlation with the change in illuminance *E*_{V}, this apparently systematic change can be explained by the changing environmental conditions which the sensors were not able to resolve. Indeed, when analyzed according to Eq. (7), such a systematic deviation cannot be confirmed. The two-color inline compensated length provides a more reliable estimate of the length.

Equation (2) is based on the assumption that both beams will follow an identical path. However, due to the large dispersion, beams of the two different colors can be expected to propagate differently. Indeed, the high time resolution data over 864 m depicted in Fig. 4(b) show traces of this effect. The signal amplitudes for the four wavelengths that have 10 *μ*s time resolution are shown. There is a significant difference in shape between the green and infrared beams, while each pair of wavelengths coincides almost perfectly. Such effects can be theoretically described.^{29} For an experimental quantification, one green wavelength and the corresponding infrared wavelength were analyzed. In Fig. 4(c), the two-color inline compensated result is shown together with the raw OPL of one 1064 nm wavelength. The two-color inline compensated result varies by more than 2 *μ*m. The raw data, however, vary only by 0.5 *μ*m. For our system, the resulting error on the order of 2 *μ*m is still negligible compared to other uncertainties. Nevertheless, this effect defines one ultimate limit of the two-color inline compensation method.

In conclusion, in this work, we demonstrated a high-accuracy, SI-traceable distance measurement system over several hundred meters. To deal with the uncontrollable environment, we chose two-color inline refractivity compensation based on synthetic wavelength interferometry. Our study stresses that, due to the unfavorable uncertainty scaling, extremely high accuracy is required for the underlying OPL observations. A standard deviation below 2 nm of the difference of the four measured OPLs was required in order to achieve sub-millimetric uncertainties under field conditions. Temporal high-resolution data provide further glimpses into the complex wave propagation that takes place under these circumstances. In the future, the two-color long-distance interferometer will serve as a primary standard in Germany for the realization of the unit length over several hundred meters.

The authors would like to thank the Finnish Geospatial Research Institute (FGI) for its hospitality at the Nummela reference baseline and the European Metrology Research Programme (EMRP) for its financial support within the JRP SIB60 Surveying. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.