We present a tracking interferometer with an intrinsic compensation of the refractive index of air. By using both wavelengths of a frequency doubled Nd:YAG laser the refractive index of air can be determined and compensated by the dispersion. One dimensional benchmark verification experiments in air conditioned and typical harsh, uncontrolled environment show an asymptotic length dependent uncertainty in the order of 0.1 μm/m for distances over 10 m, proofing the potential of this approach for high accuracy measurements in industrial environments.
© 2016 Optical Society of America
The free space measurement of distances is a fundamental challenge in precision engineering of large parts like the assembly of large wing sections in aerospace industries or the calibration of large coordinate measurement machines. State of the art commercial measurement equipment achieves relative accuracies in the order of 10−6. Recently, there has been considerable progress in the field of absolute interferometric measurement systems (e.g. [1–5]). However, no matter how sophisticated the optical measurement is designed, one still has to deal with the fact that measurements in manufacturing are typically not performed in vacuum, but in air. To make it even more challenging, typical machine shop environments are characterized by localized heat sources and restricted accessibility in the line of sight. The index of refraction, and thus the optical wavelength as inherent scale of an optical measurement, changes, however, by almost 1 ppm per Kelvin temperature change . It is immediately clear that the compensation of the index of refraction is the biggest obstacle in large scale optical metrology today , in particular in the most–critical industrial environment.
Besides the conventional approach of the application of correction formulas [6,8] based on sensor measurements, there are a number of alternative compensation approaches. For fixed and well-defined small geometries, compensation with the help of a stable artefact can be an option . For more voluminous and more flexible measurement tasks an immaterial probe is more suitable. A spectroscopic determination of the environmental parameters, e.g., gives access to the optically effective parameters along the whole beam path [10, 11]. Similarly, a measurement of the speed of sound has been proposed for the refractivity compensation . These two approaches, however, require significant extra equipment and additional measurement volume. As a solution to this problem, Earnshaw and Owens introduced already in 1967 the idea to compensate for the index of refraction by using the optical path determined with two different wavelengths and exploiting the dispersion . This approach has drawn increased attention by the length metrology community. In recent years several proof–of–principal experiments were published using frequency doubled laser sources [14–17] or frequency combs [18,19]. Recently, we proposed to use this approach to realize a primary standard for geodetic length measurement up to 1 km distance .
In this paper, we report on the successful implementation of this measurement technique into a tracking heterodyne counting interferometer, the so-called “3D–Lasermeter” within the European EMRP project “LUMINAR”, demonstrating the applicability of this technology for a typical one dimensional measurement geometry in manufacturing environments. We study the performance of this device in archetype simulations of these conditions, giving insights into capabilities of this approach, but also indications of its limitations. The intended applications are three dimensional multilateration measurements with four devices up to a distance of 20 m .
2. Experimental: 3D-Lasermeter
The optics of the 3D–Lasermeter is divided into two functional units: source and the compact movable interferometer head. The optical source is based on a frequency-doubled Nd:YAG laser (Innolight Prometheus 20), emitting at 532 nm and 1064 nm. The laser is stabilized on an iodine hyperfine structure transition at 532 nm using a frequency modulated spectroscopic set-up. Green and infrared laser beams are splitted in two paths and pass two 100 MHz and 97.5 MHz acousto-optic frequency shifters to generate heterodyne signals. The four resulting beams are coupled into polarization maintaining single mode fibers and fed into the interferometer head.
The schematic interferometer set-up is shown in Fig. 1. It is designed as a counting heterodyne two color interferometer. Two light beams are used to acquire measurement and reference beam signals and are therefore labeled “Sig”. The light acting as local oscillator to mix the signals down to the heterodyne radio frequencies is labeled by “LO”. 50 percent of the 532 nm light passes the non-polarizing beam splitters BS1 and BS2 and interferes in BS3. This is the reference path of the interferometer which measures phase changes accumulated on the path from the source via the fibers. The measurement beam is coming from BS2 and is reflected by the polarizing beam splitter PBS1. Subsequently, it passes a broadband quarter waveplate λ/4, BS4, and the beam expander. The reflected light from the triple mirror is picked off by BS4, passes an interference filter for 532 nm (IF 532 nm) and hits a position sensitive detector (PSD). The output signal of the PSD is used for tracking the center of the triple mirror. The other half of the light passes PBS1, PBS2, another broadband quarter waveplate λ/4 and is focused onto the fixed reference sphere. The light reflected from the sphere finally exits the measurement path at PBS2 and interferes with the local oscillator beam in BS5. The infrared beams are aligned on BS1 and BS2. Thus, they traverse the same path as the green beams.The interfering beams at the outputs of BS3 and BS5 each pass a corresponding interference filter for 532 nm and 1064 nm and hit the photo detectors for the reference path (RD) and measurement path (MD).
For the applications sketched in the introduction, it is important that the interferometer can be aligned flexibly in 3D, possibly following a moveable target. Around the year 2000, an optical design for such a tracking interferometer was developed by Hughes et al.  and Härtig and co–workers . It is based on the idea that the whole interferometer set-up follows the movement of a retroreflector in space. The mirror of the reference path is replaced by a sphere whose position is fixed in space. The rotatable interferometer hence measures the distance between the movable retroreflector and this fixed reference sphere [22,23]. To adapt this measurement mode, the original demonstrator based on standard opto–mechanical components  had to be miniaturized. After shrinking to an outer size of 180 × 82 × 52 mm3, the head could indeed be mounted on a base whose mechanical design is based on the commercial LaserTracer system of Etalon AG. The resulting sompact 3D–Lasermeter head is shown in Fig. 2. The main challenge was to construct a compact design while retaining the possibility to manually align four beams on their desired paths. To achieve this, the design depicted in Fig. 1 had to be changed slightly. The fiber collimators and the combining optics had to be stacked in two layers. As a benefit of this careful miniaturisation, the mechanical stability was greatly improved over the non-tracking prototype made from standard opto-mechanical components. Fortunately, the performance for refractive index compensated measurements turned out comparable.
The interferometer signals from the four photo detectors were amplified with a bandwidth of 10 MHz and digitized and processed using a 16 bit 100 MSamples/s analogue digital converter (Struck Innovative Systeme SIS3302). Onboard field programmable gate arrays (FPGA) multiply the raw signals by precalculated 2.5 MHz sine and cosine signals (1000 values) and add them up. The sum is proportional to the sine and cosine of the interferometer phase. Then the differences sin(ΦM − ΦR) and cos(ΦM − ΦR) of the phases ΦM in the measurement and ΦR in the reference pasth are derived and the fringes are counted. The resulting counter values can be read by a computer at a rate of 50 kHz, giving a rather low maximum speed of ≈25 mm/s for the 532 nm light. The limiting factors were the data rate to the computer and the limited resources on the Spartan 3 FPGAs. Meanwhile an optimized design was completed using a 250 MSamples/s converter from National Instrumens which allows a maximum speed of around 500 mm/s, which is useful for practical applications.
The phases are used to calculate the optical path lengths l1,2 for green and infrared light, which are the products of the refractive index n1,2 and the distance l. Detailed information about the equations used for calculating the refractive index compensated length l can be found in [16,24]. In brief, the equation of the refractive index from Bönsch and Potulski  can be written as13]: 24]: Eq. (3) is approximately 65.5. The drawback of the method is the fact that uncertainties in the optical path length difference l2 − l1 are therefore enlarged by this factor as well. It should be noted that although Eq. (4) does not contain such a factor explicitly, the scaling of uncertainties follows the same unfavorable behavior.
3.1. Verification under standard laboratory conditions
The 1D performance of the 3D–Lasermeter was verified at the 50 m comparator of the Physikalisch–Technische Bundesanstalt (PTB), the national metrology institute (NMI) of Germany, in a well–controlled and homogeneous environment at 20 °C. The device was compared to the Helium–Neon (HeNe) reference laser interferometer (Zygo 7112) in a folded beam path configuration up to 20 m as depicted in Fig. 3 (left). The beams, reflected from the large triple mirror, are in a lower plane compared to the beams coming from the interferometers, so there are no collisions with the small triple reflectors. This set-up almost perfectly cancels the Abbe errors due to the limited straightness of the rail. The tracking measurement mode was switched off since thermally induced movements of the rail perpendicular to the measuring axis would influence both interferometers in a different way.
For each distance, 1024 data points were acquired as 10 μs averages of the FPGA lock–in. Their mean values are depicted in Fig. 3. Each color in the diagram is assigned to a different measurement with a duration of up to nine hours. A reproducible deviation of more than 2 μm appears at a length around 2 m in the corresponding upper datasets. The dispersive refractive index compensation scales uncertainties in the difference of both lengths as measured with 532 nm and 1064 nm wavelengths by a factor A ≈ 65.5. Hence, the deviation observed in the refractivity–compensated length corresponds to a deviation between the optical distances of about 30 nm around 2 m. Indeed, following this experiment, a curve in the guiding rail of ≈ 0.4 mm was identified. The folded beam path alignment which compensates this mechanical distortion for conventional interferometers obviously fails in the case of the two–color measurement, indicating slightly different beam paths of the two wavelengths on the bench. This sensitivity was unexpected. The green and infrared beams with a 1/e2 diameter of ≈2.5 mm were aligned along the complete measurement path to match within approximately 0.3 mm. This mechanical interpretation is also supported by the fact that in tracking measurement mode, this deviation vanishes, as shown as the red dots (shifted by −2 μm) in Fig. 3. It should be noted that aside from this mechanically explainable deviation, the standard deviation of the data at short paths is 0.48 μm and increases to 0.83 μm in the range 16.5 m to 20 m. Hence, the relative length–dependent deviation remains below 10−7. It is stressed that this remarkable and reproducible performance was achieved without any other auxiliary sensor data for the refractive index compensation than humidity information.
3.2. Verification under controlled “harsh” conditions
The potential of this measurement approach in rough environmental conditions was demonstrated in a dedicated measurement campaign at the 50 m interference comparator of the Polish NMI Glowny Urzad Miar (GUM). For this purpose, two housings with 2 kW heating fans were installed from 1.8 m to 3.8 m and 5.5 m to 7.5 m along the bench. These allowed the introduction of strong temperature gradients with heavy turbulence, simulating localized strong heat sources typical for industrial environments. The environment was monitored by 38 temperature sensors along the bench with a spacing from 1 m near zero to 1.5 m at 50 m. Five hygrometers were placed evenly spaced according to the measurement lengths of the experiments and one barometer and a CO2 sensor were near the zero position. All sensors were read in a 15 s interval. The result of an experiment with the 2 kW heaters and the reflector placed on a fixed position on the bench at 19 m is shown in Fig. 4. Four temperature sensors at 2.2 m, 3.2 m, 6 m, and 6.9 m were located in the heated boxes from a total of 16 sensors up to 19 m. A folded beam configuration similar to the one described in the previous subsection was used. Again, the mean value of 1024 samples of 10.24 ms was used for each point. Unfortunately, a Helium–Neon reference interferometer could not be used in this experiment as it suffered from beam interruptions due to strong turbulence. To derive conclusions nevertheless, the distance was evaluated from the measured optical path lengths in two alternative ways: the refractive index compensated result according to Eq. (4) and the classical calculation with Edlén’s equation and the sensor data for both wavelengths. For the data in Fig. 4 the 19 m dead path was subtracted, giving only the measured length changes Δl.
The heaters were switched on at 12:17 and switched off at 12:25. Figure 4(a) shows the observed length changes calculated based on the optical path change corrected for the index of refraction using sensor data and Edlén’s equation for 532 nm in green and for 1064 nm in red color. The intrinsically refractive index compensated result is depicted in blue. Both the former “conventionally corrected” lengths apparently reduce by 30 μm due to the heating while the refractive index compensated values stay constant. The mechanical length can be assumed to be nearly constant during the experiment. It should be noted that thermal expansion due to the temperature change should lead to an increased length, contrary to the observation. One explanation for the observed increased noise is a different path for both wavelengths due to the air index gradients and dispersion. Such effects were theoretically predicted by Wijaya and Brunner , but the magnitude of the effect is still difficult to predict. In our case, the maximum observed deviation from zero of −10 μm for the compensated result (blue curve near 12:25) corresponds to a difference of 10 μm/(A = 65.5) ≈ 0.15 μm in the 19 m optical paths of both beams, giving a relative difference of 8 × 10−9l.
Figure 4(b) provides another possible explanation for the increased noise. The fluctuating signal amplitude could explain a strongly varying signal–to–noise ratio. It should also be noted that one “by–product” of the two–color system is an internal correction possibility for turbulence. The redundant information of the two optical interferometers can be used to correct for turbulence–induced wrong phase unwraps in one of the two interferometers. In the measurement shown in Fig. 4, e.g., several faulty counts (λ/2 errors) for 532 nm occurred which could be corrected using the 1064 nm data. Therefore, the two–color system can run in environments in which conventional counting interferometers fail. Excessive turbulence, however, leading to beam interruptions cannot be recovered for the counting interferometer design nevertheless.
Hence, the result of this measurement campaign clearly shows that it is possible to compensate strong temperature fluctuations with the dispersive in-situ refractive index compensation. Figure 4(c) depicts the difference Δt between the temperature measured by the installed monitoring system along the beam path and the effective “optical temperature” which can be derived from the raw length values for green and infrared, the measured humidity and air pressure according to . The average sensor temperature, starting at 18.95 °C, reaches a maximum of 19.57 °C and ends at 18.98 °C. The highest single sensor temperature was 21.58 °C in the second box at position 6 m and the sensors behind 8 m were not affected. This comparison demonstrates the advantage of this approach towards a conventional sensor–based measurement: measured temperature is up to 1.5 °C lower than the calculated one from measured dispersion, fully explaining a length error of nearly 30 μm for the 19 m path. We assume that the temperature sensors, which were 20 cm below the beam path, see a different environment than the optical beams since both heaters in the housings were placed on the floor - a problem which can never be resolved completely when using conventional sensor systems.
3.3. Verification in real industrial conditions
In March 2016 the system was studied at the AIRBUS site in Filton, UK, in a real industrial environment. As a benchmark experiment, a diagonal path of 44.8 m length was monitored for several days, the beam traversing along this path through a sensor “tunnel” made of twelve quadratic frames each carrying four thermocouples. The lowest frame was mounted at a height of ≈2.5 m and the highest one at nearly 20 m. We placed our 3D–Lasermeter on a tripod approximately 5 m apart from the lowest frame 1 and targeted a retroreflector at the end of the tunnel. Humidity and air pressure were recorded at a desk in the middle of the building. Unfortunately, the resolution of the air pressure measurement was only 1 hPa due to a software problem. To simulate different environments, the heating of the building was switched off over the weekend. Temperature dropped down to 7 °C on Monday morning with a very stable distribution. It was switched on again on Wednesday morning, introducing large spatial and temporal gradients of the temperature distribution. Length measurements along the 44.8 m path were performed continuously with the 3D–Lasermeter. However, a length reference with nanometric resolution was not available. Therefore, we decided to analyze the observed refractive index by the system to draw performance conclusions. In principle, the refractive index n can be derived by dividing the measured optical path length nl for one wavelength by the refractive index compensated result l (see ). However, measurements along the path started with the counters for both wavelength set to zero, leading to an undefined result as the absolute length cannot be determined with the 3D–Lasermeter. Fortunately, the refractive index change Δn can be derived assuming reasonable starting values. With the starting values n532–0, n1064–0, and the 44.8 m dead path ld the measured time dependent optical path length changes Δl532 and Δl1064 can be expressed as
Here Δl is the length change with time. The refractive index change Δn, e.g. for 532 nm, can be derived as
The parameters measured by the 3D–Lasermeter are the optical path length change Δl532 and Δl as the refractive index compensated result. Additional uncertainties are introduced by the sensor data used for calculating n532–0 (un532–0) and by ld (ud). Their contributions can be calculated by
With ld=44.8 m and a maximum length change of ≈ 3 mm during a measurement the term in Eq. (7) is of the order 7 × 10−5u(n532–0). Assuming a 10 °C temperature change during a measurement (Δn ≈ 10−5), the term in Eq. (8) is 2.2 × 10−7u(ld). The starting values therefore have little influence on the resulting change in refractive index due to the long path.
Figure 5 shows the result of the measurement with the largest temperature variation during heating. The refractive index change as calculated from the sensor data (temperature, pressure, and humidity) is given in red and the values from the interferometer according to Eq. (6) in black. The red curve exhibit several steps of ≈ 2.7 × 10−7 due to the limited resolution of the pressure. The strong turbulence at the end of the measurement induces beam interruptions which lead to non-recoverable jumps in the data (see vertical arrow in Fig. 5). Overall, however, the difference between sensor and interferometer data (magenta curve) is below 5 × 10−7 as indicated in the graph.
It is instructive to derive the effective temperature from the optical path lengths, pressure and humidity information. Figure 6(a) shows this temperature in black together with the temperature from sensor data in red. Since the starting value of the temperature is used for the dead path as discussed above, the first temperature from the interferometer is identical to this value. The difference between both temperatures is within 0.5 K until fringe counting errors occur. It is obvious, however, that the temperature derived from the sensor data is systematically higher. A possible explanation is the temperature distribution in the building, which is given in Fig. 6(b). The x-axis denotes the frame number, from 1 at the lower end (height ≈2.5 m) to 12 at the upper end near the reflector (height ≈20 m). During heating a large temperature gradient of nearly 10 K builds up with the steepest increase at the lower end. Since the first 5 m of the measured path has no thermometer, the interferometer measures a somewhat lower effective temperature.
3.4. Uncertainty budget
It is instructive to investigate the achievable combined measurement uncertainty of the 3D–Lasermeter. In this section, it is derived following the “Guide to the expression of uncertainty in measurement” . The contributions are summarized in Table 1. In general, the uncertainty budget is dominated by in principle small uncertainty components that are scaled up by the A factor of 65.5. This is elaborated in more detail in the following.
In a first step, length–dependent contributions are discussed. As intended by design, refractivity–associated uncertainties remain well below 10−7l. Three uncertainty contributions are associated with refractivity. For the compensation according to Eq. (4), the partial pressure of water vapor pw has to be determined. Usually, the relative humidity RH is measured primarily. The partial pressure of water can then be deduced if the temperature is known. Although the partial pressure depends exponentially on the temperature, the effect on the uncertainty has only little influence in our case so that an uncertainty of about 1° C has only little influence. In reasonable conditions, an absolute uncertainty of 2% RH of the relative humidity measurement can be expected. According to Table 1, this measurement still leads to the second largest length–dependent contribution.
The uncertainty of the used Edlén equation in the modification by  is given by Bönsch and Potulski as 1 × 10−8. This contribution influences the interpretation of the optical path lengths also when using Eq. (4). According to Eqs. (2) and (4) the uncertainties in the difference between them are scaled by A. However, the resulting length uncertainty depends on whether the deviation from the model for the two wavelengths has the same or the opposite sign. If the deviation goes into the same direction for both wavelengths, the differencing in Eq. (2) cancels this effect almost perfectly. In case of opposite deviations, however, this would lead to a systematic uncertainty contribution up to 6.5 × 10−7l. The experimental results at PTB and GUM indicate that there is no scaling of uncertainties to this amount. Thus, we assume that systematic antisymmetric deviations of the model from the real refractivity can be excluded for the two wavelengths involved. To account for the general uncertainty of the Edlén model of , we assign this uncertainty to the functions K(λ) and g(λ) in Eq. (1), leading to a negligible contribution to the final length result.
The most critical length–dependent uncertainty contribution for the 3D–Lasermeter can be attributed to the cosine error between the green and the infrared beams. According to the optical design given in Fig. 2, they have to be manually superimposed using the wedged plates. For this purpose, an optical path of 40 m was used. Nevertheless, due to self diffraction, the beam waist diameter increases from initially 2.5 mm to approximately 10 mm (green) and 20 mm (infrared) over this 40 m distance. This limits the resolution of this procedure. In addition, different wavefront propagation for the two colors over longer distances can lead to cosine–like behavior as well. First verification experiments showed a remnant cosine error in the order of 10−8l. Fortunately, this can be corrected by one order of magnitude using an additional calibration factor. It can be determined on an interference comparator. Green and infrared interferometer are evaluated independently on the same distance for this procedure, applying a sensor–based refractivity compensation. As can be seen in Table 1, the uncertainty due to the cosine error could thus be reduced to 10−9l. This can be attributed to the mechanical stability of the set–up. Due to the uncertainty scaling of 65.5, however, this still translates into an uncertainty contribution in the order of 3.78 × 10−8l.
The constant component of the combined measurement uncertainty can be mainly attributed to mechanical and optical origin. The motors of the tracking mechanism introduce vibrations in the interferometer which are slightly different for both wavelengths. For the modeling of the effect, it makes sense to divide them in two parts: synchronous vibrations with σ = 110 nm standard deviation affect the common beam paths in the interferometer (not scaled by A) and asynchronous vibrations, which appear as the difference between the lengths from green and infrared. They are small (σ = 3 nm) but are scaled by A. During the 1D measurements the tracking was switched off, but the motors remained powered. The vibrations were somewhat lower in this case.
Another contribution to the constant component can be associated with drifts in the length difference measured with both wavelength over time. Stability measurements on a short path showed drifts of the length difference between green and infrared results of up to 12 nm in an air conditioned lab for fixed identical lengths. During temperature changes we observe periodic variations of the length difference. We attribute these drifts to optical effects. We have identified temperature and stress–induced polarization variations in the beam splitters as most probable explanation. An optimal technical solution for this problem has not been found yet. Unfortunately, these drifts are scaled by A and are the dominant part of the constant uncertainty component.
It is interesting to note that “classical” sources of uncertainty in interferometry play a minor role for the length uncertainty in comparison with the previous contributions. The uncertainty of the vacuum wavelength itself of 10−9λ results in a negligible length–dependent uncertainty of below 10−9l. The phase noise is estimated to 1.18 × 10−3 radians, corresponding to 0.1 nm for green and 0.2 nm for infrared. Although this contribution is also scaled by A, it has a minor influence below 10 nm on the overall length measurement uncertainty.
When applied for 3D measurements in space additional length–independent contributions from the beam steering mechanics and a slight parallel shift between green and infrared beams have to be taken into account. Recent investigations on a coordinate measurement machine indicate that the constant part of the uncertainty is 1.3 μm for fixed temperature up to estimated 3.8 μm for a 5 K temperature change during the measurement due to thermal expansion. The length dependent part is not affected.
In conclusion, the extensive characterization experiments in different benchmark environments have proven the capabilities and the potential of the dispersive in-situ refractivity compensation of the 3D–Lasermeter. Temperature does not need to be known for calculation of the length, only a humidity sensor is necessary. The 1D performance was demonstrated in comparison with a HeNe interferometer. The length-dependent uncertainty remains below 10−7l, but the immanent unfavorable scaling of uncertainties by the large factor A ≈ 65.5 leads to a quite high constant part of 1 μm. Under harsh conditions with strong turbulence the measured length is only affected by a higher noise level. Free space measurements over a 44.8 m path with temperature gradients up to 10 K were consistent within 5 × 10−7 to conventional sensor data. The systematic deviation is well explicable by an imperfect sensor alignment.
For both experiments under harsh conditions, one could argue that a more sophisticated reference sensor design would have led to a better classical compensation result. But the networks were already designed beyond good practice in large scale metrology applications in industry. More importantly, no matter how hard one tries, the temperatures captured by sensor data located only in the vicinity of the beam substantially deviates from the effective value. In practice this leads to a typical accuracy limit of 1 ppm in large–scale applications. It should also be noted that for application, the costs for the installation maintenance of sensor networks rise significantly with the sensor density. Optical–refractivity compensation, though non–trivial to implement, overcomes this problem. By these experiments, we demonstrated that a field–capable realization of this measurement principle is possible. Due to its movable interferometer head, the 3D–Lasermeter enables high accuracy measurement of large structures in uncontrolled environments, using, e.g., measurement strategies as proposed by Wendt et al. .
The authors would like to acknowledge funding from EMRP IND53 “LUMINAR” of the European Metrology Research Programme (EMRP). The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.
The authors would like to thank for the provision of the reference bench by GUM (D. Czulek, R. Szumski, M. Wisniewski), the friendly hospitality of E. Szigeti and M. Loparco from AIRBUS, and A. Lewis from NPL and L. MacDonald from UCL for the environmental reference data of the AIRBUS experiment. The authors also acknowledge fruitful discussions with W. Schott, P. Köchert, K. Rost, K. Wendt and F. Härtig as well as technical support from J. Bautsch. The authors also appreciate the valuable comments of the two anonymous referees.
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