## Abstract

We present an implementation of an absolute distance measurement system which uses frequency scanning interferometry (FSI). The technique, referred to as dynamic FSI, uses two frequency scanning lasers, a gas absorption cell and a reference interferometer to determine the unknown optical path length difference (OPD) of one or many measurement interferometers. The gas absorption cell is the length reference for the measurement system and is traceable to international standards through knowledge of the frequencies of its absorption features. The OPD of the measurement interferometers can vary during the measurement and the variation is measured at the sampling rate of the system (2.77 MHz in the system described here). The system is shown to measure distances from 0.2 m to 20 m with a combined relative uncertainty of 0.41 × 10^{−6} at the two sigma level (*k* = 2). It will be shown that within a scan the change in OPD of the measurement interferometer can be determined to a resolution of 40 nm.

© 2014 Optical Society of America

## 1. Introduction

Absolute distance measurement systems, as opposed to the more common displacement measurement systems, which can be used for distance measurements up to several tens of meters, with relative uncertainties of less than 1 × 10^{−6}, are of significant interest in the field of metrology, especially in the manufacturing industry and big science projects. Measurement systems with these capabilities could directly improve the manufacturing efficiency and accuracy of large assemblies. For large airplane wings this could lead to reduced reworking through interchangeable rivet hole patterns requiring 40 *μ*m accuracies on hole positions across the wing as well as less fuel consumption by enabling the production of natural laminar flow wings which are expected to require profile tolerances of approximately 0.5 mm across 20 m to 30 m long wings. The implementation of tooling that can achieve this for a dense set of point across the wings would require ppm scale accuracy in absolute distance measurements. Other applications would lie in particle accelerators and colliders enabling tighter alignment tolerances and in turn more complex and advanced designs. There are many different techniques for absolute distance measurement using lasers, based upon: time of flight, synthetic wavelengths or Frequency Scanning Interferometry (FSI) [1, 2].

The realisable accuracy benefits of FSI require careful control of several potential sources of uncertainty [3] and provision of a suitable length reference. Many of the FSI approaches utilised to date have used conventional (air spaced) optics and have relied on using a reference interferometer which is required to be both stable during the FSI scan and possess longer term stability to provide the length scale traceability for the system [3].

Moving away from the use of stable reference interferometers has seen the use of stable frequency references in the form of additional stabilized lasers operating as single wavelength fringe-counting interferometers [4] and synthetic wavelength systems [5]. These have, in turn been superseded by the use of frequency combs [6, 7] operating at very high levels of accuracy (up to 1 × 10^{−8}) and fast data rates (kHz). However, many of these systems provide only one channel of measurement despite complex optical arrangements, including some which use highly vibration and temperature sensitive components, precluding their use in industrial settings.

There are many situations where it is not only important to measure distances with relative uncertainties at the 1 × 10^{−6} level, but multiple distances with this uncertainty simultaneously. Examples in manufacturing are the simultaneous measurement of machine tools or CMM probes from multiple measuring stations to determine the position and attitude of the tool.

In this paper we will present our latest advances in the area of FSI. The developments we discuss improve on previous implementations in the areas of reference interferometers and traceability to the unit of length. In previous implementations the Optical Path length Difference (OPD) of a reference interferometer was the primary length scale and thus required to be calibrated and stable stable over long periods of time. This lead to complex, expensive and bulky implementations [8]. These problems are overcome here through the use of a gas absorption cell. It will be shown that the gas absorption cell allows the length of the reference interferometer to be evaluated during each laser scan, therefore removing the requirement to have an OPD which is stable over the calibration lifetime. As the OPD of the reference interferometer is measured via the gas cell during each measurement, the gas cell becomes the length standard for the system. The use of the gas absorption cell has also simplified the analysis process. As the frequencies of the absorption features of a gas cell are fundamental to the gas species within the cell, they are very stable. The frequencies of the features are known at different gas pressures as they have been characterised by national measurement institutes (NMIs) such as the National Institute of Standards and Technology (NIST) or the National Physical Laboratory (NPL). We are using a Doppler broadened absorption cell to keep the system inexpensive and the technique simple. These properties make the gas cell ideal for the length standard as it is easily traceable to the international standards realised at these, or other, NMIs. As the length standard for the system is now the gas cell, the gas cell acts as the traceability link to international standards (SI) through the speed of light.

The system described here is easily scalable allowing one core system to support a large number of measurement interferometers simultaneously. The number of measurement interferometers is limited by the power of the lasers, which can be readily increased through the use of Erbium Doped Fiber Amplifiers (EDFAs), and the amount of computing power available for data analysis. The measurement interferometers are also low cost in comparison to the core parts of the system. The system described here is robust, compact and readily deployable into many environments.

The underlying theory will be described in section 2 with the calibration experiment and results shown in section 3. The effect of thermal changes on the gas cell and how they can be corrected for are shown in section 4 and a summary of the measurement process and uncertainties are described in section 5. Measurements covering a range of distances up to 20 m are shown in section 6 with the conclusion in section 7.

## 2. Theory

#### 2.1. Basic frequency scanning interferometry (FSI)

Basic FSI [3, 8–10] is an absolute distance measurement technique which uses a frequency scanning laser to measure the ratio of the optical path length differences (OPD) of two interferometers. If the OPD of one of the two interferometers is known then the OPD of the second interferometer can be determined. The interferometer with the known OPD is referred to as the reference interferometer and has an OPD which is assumed constant over long periods. The interferometer with the unknown OPD is referred to as the measurement interferometer and its OPD in the basic FSI technique is also assumed to be constant during a scan. The basic FSI setup is shown in Fig. 1. It should be noted that Fig. 1 shows only two interferometers connected to the system, but in general many interferometers can be connected, limited only by the power of the laser.

The interferometers used during the experiments are fiber fed Fizeau interferometers. A Fizeau interferometer has a zero length reference arm and therefore the OPD is twice the interferometer optical length (labelled L_{R} and L_{m} in Fig. 1). Future discussions will refer to the optical length of the interferometer (or the length) not the OPD. All of the interferometers are connected to the frequency scanning laser via a tree of fused taper fibre splitters.

A measurement is made as follows: The laser sweeps its frequency from a start frequency (*ν*_{t0}) to an end frequency (*ν*_{tn}), with the intensity of both interferometer outputs recorded and digitised. The recording is referred to as a scan and is digitised at discrete times *t _{i}*. The output intensities of the interferometers vary sinusoidally with laser frequency and the absolute phase of the sinusoidal function generated by the reference interferometer given by:

*ϕ*

_{abs ti,R}is the absolute phase of the reference interferometer at time

*t*,

_{i}*L*is the length of the reference interferometer,

_{R}*ν*

_{ti}is the frequency of the laser at time

*t*and

_{i}*c*is the speed of light. The absolute phase cannot be extracted from the intensity measurements without additional information. However, the phase change relative to some time,

*t*

_{0}, can be extracted. The relative phase will be referred to as the extracted phase throughout the paper. The phase is extracted from the interferometer intensities using a Hilbert transform [11–13]. The Hilbert transform generates a sine wave signal from a cosine signal; the inverse tangent of the ratio of the signals then gives the wrapped phase. The wrapped phase has phase values between −

*π*and

*π*which are then unwrapped into a continuous phase to get the extracted phase. The extracted phase of the reference interferometer is given by:

*ϕ*

_{ti,R}is the reference interferometer extracted phase at time

*t*, and

_{i}*ν*

_{t0}is the frequency at the start of the scan. The extracted phase of the measurement interferometer is likewise given by: Where

*ϕ*

_{ti,M}is the measurement interferometer extracted phase at time

*t*, and

_{i}*L*is the length of the measurement interferometer. The ratio of the extracted phases described in Eqs. (2) and (3) is equal to the ratio of the lengths: Therefore, if the lengths of the measurement and reference interferometers are constant during the scan, and the length of the reference interferometer is known, the length of the measurement interferometer can be determined. Details of implementations on how this analysis can be performed are well described in [8–10]. When the measurement interferometer is operating in air the measured length needs to be corrected for the effect of the refractive index of air to give the geometric length. The lasers used here have nominal wavelengths of around 1550 nm and so the Ciddor equation is used to calculate the refractive index of air using measured values of temperature, pressure and humidity [14].

_{M}#### 2.2. Gas absorption cell and reference length measurement

A gas absorption cell (a glass cell containing a gas such as hydrogen cyanide (H^{13}C^{14}N)) has absorption features at known laser frequencies [15]. A small amount of light from the frequency scanned laser is passed through the gas cell during the scan and the transmitted intensity is digitised. The power of the laser is also digitised and used to normalise the transmitted intensity to remove laser power fluctuations. By fitting the measured absorption spectrum features against the extracted reference phase, the extracted reference phase at the centroid of each feature can be determined. Gaussian distributions are fitted to the features using Minuit2 [16], a numerical minimisation program.

The phase extracted from the reference interferometer is given by Eq. (2). If *L _{R}* is constant throughout the scan,

*L*can be determined by fitting a straight line to a plot of the known frequency of each fitted absorption feature versus the reference interferometer phase at the centre of each absorption feature. The gradient of the straight line fit to the frequency versus phase plot is $\frac{4\pi}{c}{L}_{R}$ and therefore the length of the interferometer

_{R}*L*can be determined.

_{R}In practice any gas absorption cell can be used as long as it has sufficiently well known absorption features in the frequency range of the frequency scanning lasers. In the work described here we will use two different length hydrogen cyanide (H^{13}C^{14}N) gas cells, one of length 165 mm and one of length 50 mm. Both of the gas absorption cells have a nominal gas pressure of 25 Torr (3.333 kPa) at 25 °C. The properties of the gas cells, including the frequencies of the absorption features at different pressures and the effect of environmental variables, are described in [15].

#### 2.3. Dynamic FSI

In section 2.1 the principle of basic FSI was outlined using a single frequency scanning laser. This technique has the limitations that the lengths of the measurement interferometers must be constant during the scan, and the length of the reference interferometer must be known prior to the scan. By introducing a gas absorption cell and a second frequency scanning laser, which has its light simultaneously injected into the interferometers, the length of the measurement interferometer is no longer required to be stable. Figure 2 shows the layout of an FSI system using two frequency scanning lasers and a gas absorption cell.

The inclusion of the second frequency scanning laser allows the implementation of the technique we call dynamic FSI [13,17]. Dynamic FSI enables the determination of the length of the measurement interferometer at each digitised time point *t _{i}*. By including a second frequency scanning laser a second interference pattern is generated for both interferometers. The two lasers tune at different speeds, in opposite directions and over different, but overlapping, laser frequency ranges. However, the lasers do scan simultaneously and for the same length of time. As the lasers tune at different speeds they produce intensity oscillations of different frequencies (not to be confused with the laser light spectrum) and can therefore be separated after digitisation using Butterworth bandpass filters [18] in the frequency domain. The phase for each laser signal is then extracted using a Hilbert transform as described above. The second interference pattern generates a second phase equation for each interferometer and requires a change in notation for Eqs. (2) and (3). The frequency of

*laser 1*is now represented by

*ν*

_{ti,1}and that of

*laser 2*by

*ν*

_{ti,2}. The measurement interferometer’s length is now

*L*

_{ti,M}and can vary during the scan. The extracted phase of the reference interferometer is now

*ϕ*

_{ti,R,1}for

*laser 1*and

*ϕ*

_{ti,R,2}for

*laser 2*. Similarly the extracted phase of the measurement interferometer is

*ϕ*

_{ti,M,1}for

*laser 1*and

*ϕ*

_{ti,M,2}for for

*laser 2*.

The dynamic FSI implementation described in [13] requires that the reference interferometer length be known prior to the scan, and the analysis has to perform a non-linear least squares fit to determine the measurement interferometers’ lengths. These requirements can be removed with the introduction of a gas absorption cell. The light from both lasers is passed through the gas absorption cell generating two spectra, one for each laser, and the reference length for each spectrum can be determined as described in section 2.2. As the gas absorption cell has absorption features at known frequencies, new equations for the extracted phase can be determined. The same absorption feature is selected in both gas cell spectra and used to determine the frequency of the corresponding laser at that point in the scan. The time at which *laser 1* is at this known frequency is *t*_{a1} and the time *laser 2* is at the frequency is *t*_{a2}. A third time *t*_{a3} is also defined near the middle of the scan. *t*_{a3} can in principle be defined at any point in the scan but for reasons of numerical accuracy it is most convenient to select a point near the middle of the scan.

The extracted phase from the reference and measurement interferometers, for *laser 1* and *laser 2*, can now be written as:

*t*

_{a1},

*t*

_{a2}and

*t*

_{a3}is given by:

*L*is known (see section 2.2) then Eqs. (13) and (14) can be used to determine

_{R}*ν*

_{ti,1}and

*ν*

_{ti,2}for all times in the scan. This includes the determination of

*ν*

_{ta3,1}and

*ν*

_{ta3,1}which are frequencies to arbitrarily chosen time indices. With

*ν*

_{ti,1}and

*ν*

_{ti,2}determined the only two unknowns remaining in Eqs. (15) and (16) are

*L*

_{ta3,M}and

*L*

_{ti,M}. The value of

*L*

_{ta3,M}can be determined by dividing Eq. (15) by Eq. (16) to give:

*L*

_{ta3,M}cannot be determined from measurements at time

*t*

_{a3}, and it is difficult numerically for times close to

*t*

_{a3}, therefore

*L*

_{ta3,M}is calculated as an average from data points at either end of the scan. Once

*L*

_{ta3,M}has been determined, Eqs. (15) and (16) can be used individually to determine

*L*at each point in the scan. Any difference in the values of

_{i,M}*L*as determined from Eq. (15) and Eq. (16) are due to errors in the phase extraction of the reference and measurement interferometers or the frequency determination for both lasers. The two measurements of

_{i,M}*L*from the two lasers can be averaged to get a single measurement of

_{i,M}*L*.

_{i,M}## 3. Gas cell calibration

Two gas absorption cells are filled with hydrogen cyanide (H^{13}C^{14}N) at a nominal pressure of 25 Torr (3.333 kPa). One gas cell is short (50 mm in length) the other is long (165 mm in length). The longer gas cell gives deeper and broader absorption features in comparison to the shorter. The data sheets for the gas cells [15] give the frequency of the absorption features, along with their uncertainties, at different gas cell pressures. To determine the length of the reference interferometer the absolute frequency of each of the gas cell absorption features is not required, what is required are the frequency differences between the features. The uncertainties of the absolute absorption feature frequencies are given in the data sheets and we suspect they may include a correlated term. If the uncertainties were uncorrelated we would expect to measure a Type A uncertainty [19] on the FSI length measurement of approximately 5 × 10^{−6} (*k* = 2) where *k* = 2 refers to a two sigma confidence level in the assumption of a Gaussian probability distribution. Note that a Type A uncertainty is often referred to as a statistical uncertainty and is defined as “evaluation of a component of a measurement uncertainty by statistical analysis of measured quantity values obtained under defined measurement conditions”. Hereafter, unless other wise stated, we will refer to this type of uncertainty as the statistical uncertainty. We will show at the end of this section that our FSI system can measure a length with an expanded statistical uncertainty of 0.32 × 10^{−6} (*k* = 2), indicating that there is a significant correlated term in the uncertainties of the frequencies of the gas cell absorption features, as expected. An example of the H^{13}C^{14}N spectrum versus extracted reference phase is shown in Fig. 3. The fit to the fitted peak phases versus the peak frequencies is shown in Fig. 4.

The pressure of the gas inside our gas cell is known only to approximately 10 %. This uncertainty in the gas cell pressure leads to a relative uncertainty of 2.5 × 10^{−6} (*k* = 2) in the measured length from the uncorrected pressure shifts in the absorption features. The uncertainty induced by the unknown gas cell pressure can be reduced through a calibration procedure. The calibration procedure does not determine the gas cell pressure or absolute frequencies but instead determines a single calibration constant by which the measured lengths have to be scaled. This practical approach circumvents the much harder problem of calibrating the absolute frequencies of many features and instead uses the more readily available wavelength of a single laser in a dedicated calibration experiment. It should be noted that in this practical approach the wavelength of this single laser becomes part of the traceability chain. It would have been possible to have the gas cell frequencies determined at a given temperature by a national measurement institute such as NPL, but this was not done as part of this work. As the calibration constant is linked to a specific gas cell, it will be referred to as the gas cell calibration constant. The temperature at which the gas cell is calibrated is also recorded to allow for correction of temperature induced pressure shifts during measurements as described in section 4.

#### 3.1. Gas cell calibration experimental set-up

The dynamic FSI set-up shown in Fig. 2 was connected to three interferometers. The interferometers are single mode fiber fed with the short arm reflection coming from the tip of the fiber which is connected to a collimator as shown in Fig. 5. The long arm reflection is from either a sphere mounted retro-reflector (SMR) or a plane mirror. All of the interferometers are connected to the two frequency scanning lasers. The lasers are external cavity diode lasers with a mode-hop free tuning range from 1520 nm to 1630nm at tuning speeds from 2 nm s^{−1} to 2000 nm s^{−1} providing up to 8 mW of fibre coupled power. The manufacturer does not quote a line width or coherence length for these lasers but we estimate the coherence length to be in excess of 10 m. Each interferometer is read out by a separate photodiode, the signals of which are amplified and digitised by custom built analogue to digital converters (ADCs). All ADC channels sample simultaneously at 2.77 MHz and with 14 bit resolution.

During the calibration the retro-reflector of one interferometer is mounted on a motion stage and the length of this interferometer can be adjusted over a wide range. The interferometer on the motion stage will be regarded as the measurement interferometer. The other two interferometers are stable interferometers which use low expansion (Invar) structures to maintain a fixed reference length (hereafter referred to as Invar interferometers). The Invar interferometers act as two independent reference interferometers. The measurement interferometer’s length is measured against both Invar interferometers. The Invar interferometers are modified versions of the Invar interferometers described in chapter 5 of reference [8]. The Invar interferometers are shown under construction in Fig. 6. The Invar interferometers are highly stable and vacuum tight but were not evacuated but sealed and air filled. The launch collimators of the Invar interferometers have been changed to the design described above.

The calibration constant for a gas cell can be determined by using the dynamic FSI system to measure the displacement of the motion stage. The displacement is also measured by an external witness. The external witness was an Etalon AG manufactured LaserTRACER which uses a Michelson style interferometer [20] with a stabilised HeNe laser. The experiment is set up so that the infra-red FSI laser is combined with the LaserTRACER’s HeNe laser via a dichroic beam splitter as shown in Fig. 7. The dichroic beam splitter is adjusted until the HeNe and the infra-red laser beams are co-linear. The HeNe and the infra-red laser beams are aligned with the motion stage across its entire motion range. Both the FSI and the LaserTRACER measure the stage simultaneously. The stage has a motion range of two meters. The stage is moved back and forth along its entire length three times and is stationary during the FSI measurements. The LaserTRACER measures the displacement of the stage continuously and thus is able to give a relative position of the stage at any time. The measurements are taken at six positions along the travel of the stage. The laser parameters used in the calibration experiment are shown in table 1. The experiment is repeated a number of times separately for both the long and the short H^{13}C^{14}N cell. A straight line is fitted to the displacements measured by the LaserTRACER versus the lengths measured by dynamic FSI. The gradient of the fit is the gas cell calibration constant. The fit takes account of the uncertainties of both the dynamic FSI and the LaserTRACER measurements, described next, to determine the calibration constant and its uncertainty.

In the mode of operation used in this work the LaserTRACER has a relative measurement uncertainty of 0.17 × 10^{−6} (*k* = 2) [21]. The LaserTRACER uncertainty includes the uncertainty from the laser wavelength, interpolation between fringes, environmental measurements (air pressure, temperature and humidity) and vibrations. The residual of a straight line fit to the distances measured by the LaserTRACER versus time (for one stage position) can be used as an estimate for the statistical uncertainty of the LaserTRACER. The straight line fit removes any linear distance changes with time. The one sigma spread of the residual for all the stage positions for one calibration experiment is shown in Fig. 8. In all calibration experiments we find the highest one sigma statistical uncertainty of the LaserTRACER is < 0.035 *μ*m.

The FSI length measurements are corrected for the refractive index of air using the Ciddor equation [14]. Assuming a temperature uncertainty of 0.2 °C (*k* = 2), a pressure uncertainty of 0.5 hPa (*k* = 2) and a relative humidity uncertainty of 2.0 % (*k* = 2) the relative uncertainty of an FSI length measurement due to the Ciddor equation is 0.22 × 10^{−6} (*k* = 2). The wavelength used in the Cidor equation is the middle laser wavelength during the scan. The laser wavelength changes during the scan and, as the refractive index depends on wavelength, the correction also changes during the scan. The effect of the changing refractive index on the measured length is at the 1 × 10^{−9} level and so is neglected here. The relative statistical uncertainty for the FSI measurement can be determined from the calibration experiment by dividing the horizontal residual of the fit by the FSI length measured. This is described in more detail in section 3.2 and found to be 0.32 × 10^{−6} (*k* = 2). The combined uncertainty (neglecting the calibration constant term) is therefore the quadrature sum of the Ciddor term and the statistical term which is 0.39 × 10^{−6} (*k* = 2).

#### 3.2. Gas cell calibration experimental results

Figure 9 shows the results of one of the calibrations of the long H^{13}C^{14}N cell. A straight line fit to the displacement vs. FSI distance measurements is used to determine the gradient, which is the gas cell calibration constant, of 0.99999810±0.12 × 10^{−6} (*k* = 2). The residual plot (middle) shows that the alignment of the FSI laser beam, the LaserTRACER beam and the motion stage is very good as the residual is centred around zero and has no systematic trend (the residual is the horizontal residual from the fit in the upper plot). The middle plot in Fig. 9 also shows that the variance of the residual increases with distance. To determine the relative residual, the residual is divided by the distance measured by the dynamic FSI. The relative residual is shown in the lower plot of Fig. 9. A histogram of the residual per meter with a Gaussian curve fitted is shown in Fig. 10 and is a measure of the relative statistical uncertainty of the dynamic FSI. The Gaussian fit to the histogram shows a relative statistical uncertainty of 0.30 × 10^{−6} (*k* = 2).

The calibration experiment was repeated a number of times for both the long and the short H^{13}C^{14}N gas absorption cells, with the results of the calibrations shown in Figs. 11 and 12. Figures 11 and 12 show the calibration constant determined using both Invar interferometers (red and blue curves) as well as the uncertainty for each calibration experiment (error bars), from the straight line fits, along with the weighted average of the calibration constants and the combined uncertainty for the calibration constant (labelled Mean and Standard deviation in text inserts). The laboratory was kept at a constant temperature of 21 ±0.5 °C during the calibrations.

Figures 11 and 12 show that the calibration experiment is repeatable as the one sigma spreads of the calibration constants, 4.3 × 10^{−8} and 5.9 × 10^{−8} using the first reference interferometer for the long and short cell respectively and 4.0 × 10^{−8} and 5.4 × 10^{−8} using the second, are compatible with the uncertainties of the gas cell calibration constants determined from the straight line fits to the displacement vs FSI data. Figures 11 and 12 also show that the calibration constants determined using the two independent reference interferometers are compatible with each other.

Table 2 shows the weighted average (over both Invar interferometers) calibration constants for the long and short H^{13}C^{14}N cells and their uncertainties. The table also shows the largest statistical relative uncertainty observed during any of the calibration experiments, the value shown corresponds to the broadest of all histograms similar to that shown in Fig. 10 and represents the measured statistical uncertainty from an FSI length measurement.

#### 3.3. Sub-scan length measurement

The dynamic FSI system is capable of measuring the length of the measurement interferometer at each sample point. This allows for the length of the measurement interferometer to vary during the measurement, it may for example oscillate or drift.

As a qualitative example of the sub-scan capability, a measurement interferometer was set-up with the retro-reflector placed on a piezo electric actuator driven by a signal generator and nonlinear amplifier, oscillating at aproximately 300 Hz. The upper plot in Fig. 13 shows the length of the oscillating OPD measured by an FSI scan. The actuator’s base oscillation frequency is clearly visible in the Fourier transform shown in the lower plot of Fig. 13. The actuator does not oscillate as a perfect sine wave as indicated by the appearance of multiple peaks representing different harmonic frequencies as anticipated for this non-linear system.

A second measurement interferometer was set-up with the retro-reflector mounted on a linear motion stage. Figure 14 upper left shows a series of twenty FSI measurements of the second interferometer at approximately two second intervals. The motion stage starts stationary, then moves its full length, remains stationary for a time and then moves back to the start. The dynamic FSI system could successfully measure the length during the 18 mm s^{−1} motion speed period shown in Fig. 14 upper right. It has also resolved the time-non-linear motion of the stage as shown in Fig. 14 lower left and right. It can be seen that the stage oscillates around a time-linear trajectory with an amplitude of approximately 1 *μ*m and a period of approximately 1.1 ms. This micrometre-level oscillation is well above any possible cyclical error of our system (c.f. Fig. 12).

These two measurements are typical applications in vibration and dynamic motion analysis and take only a few seconds (minutes for the stage measurement) to perform.

To study the sub-scan noise a dynamic FSI scan is taken using the two Invar interferometers. One Invar interferometer acts as the reference interferometer and the other as the measurement interferometer. Figure 15 shows the lengths measured during the scan which indicates that sub-scan length measurement has a peak to peak noise of approximately 40 nm. The noise oscillations in Fig. 15 are thought to be caused by phase extraction errors.

## 4. Gas cell temperature variation

During system calibration, described in section 3, the system was in a temperature controlled laboratory and the gas cell remained at a constant temperature during the calibrations. In a realistic use scenario the temperature of the gas cell will change and therefore the gas cell pressure will vary. The frequency of the gas cell absorption features are weakly pressure dependant. These pressure shifts have been measured by NIST for absorption peaks used here [15]. An uncorrected pressure shift would lead to an increased uncertainty in the measurement. To determine the pressure in these low pressure gas cells, the ideal gas law can be used to predict the pressure change by measuring the temperature. The ideal gas law states:

where*P*is the absolute pressure of the gas,

*V*is the volume of the gas,

*n*is the amount of gas (measured in moles),

*R*is the ideal gas constant and

*T*is the absolute temperature. At the time of calibration the default gas cell pressure (

*P*

_{0}) and the temperature of the gas cell (

*T*

_{0}) are known and therefore: If we measure a temperature

*T*, assuming the change in volume of the glass container to be negligible, then the pressure

*P*is given by: This pressure can then be used to determine the frequency shifts of the absorption features. We would expect to see a pressure change of 11.20 Pa K

^{−1}for a 25 Torr (3.333 kPa) gas cell. Using the pressure shifts measured by NIST [15] Monte Carlo analysis was performed to determine the measured length error generated by a temperature change to be 0.076 × 10

^{−6}K

^{−1}. For example, if a 1 m interferometer were measured, without temperature correction, if the temperature of the gas cell was one degree different from the calibration temperature we would get the measurement wrong by 0.076

*μ*m.

To verify the pressure shift correction method an experiment was set-up which includes two independent gas cells. Both gas cells are connected to the same pair of lasers. One gas cell system is placed in an environmental chamber and its temperature varied from 10 °C to 30 °C, the other is held at a constant temperature. The length of a 3.2 m long Invar interferometer is measured by both gas cell systems using the same reference interferometer and frequency scanning lasers. The gas cell in the environmental chamber is analysed once with and once without pressure shift correction. The upper plot of Fig. 16 shows the relative difference in length as measured by the two gas cell systems; the blue points include thermal corrections and the red points do not. The temperature is shown in the lower plot of Fig. 16. A histogram of the relative difference with the correction turned on is shown in Fig. 17.

From Figs. 16 and 17 it can be seen that the use of a thermal correction correctly adjusts the measurements. From the histogram in Fig. 17 the two sigma spread is 0.45 × 10^{−6} (*k* = 2) which is in agreement with what we would expect from combining the uncertainties of two dynamic FSI measurements in quadrature.

## 5. Measurement summary

To perform a measurement the intensity for the required interferometers, gas cells and laser power along with the atmospheric conditions and the temperature of the gas cell are measured. The gas cell temperature is used to apply pressure shift corrections for the gas cell as described in section 4. The phase is extracted from all of the interferometers for each of the lasers and the phase and frequency of each of the required absorption features determined. Now the length of the measurement interferometer can be determined as described in section 2. The length measured by the FSI system now needs to be corrected for the refractive index of air and the gas cell calibration constants as shown in Eq. (21).

*C*with the results shown in table 2.

_{GasCell}*C*is calculated from the Ciddor equation and the measured temperature, pressure and humidity of the air.

_{ciddor}The corrected length measurement, *L _{corrected}*, has four components to its combined uncertainty which are: the relative gas cell pressure correction uncertainty, the relative statistical uncertainty of

*L*, the relative uncertainty of the refractive index from the application of the Ciddor equation and the relative uncertainty from the gas cell calibration constant. Table 3 summarises how the refractive index error is made up from errors on the measured atmospheric parameters for the case of the calibration experiment performed here. Atmospheric errors have to be re-evaluated for each experiment. For long baselines in unprotected air it is particularly difficult to achieve temperature errors that are as small as those used in table 3. Table 4 and table 5 show the remaining uncertainties along with the combined uncertainty for a system using either the long or the short H

_{measured}^{13}C

^{14}N gas cell. For example, using the long gas cell to measure a distance of 1 m would result in a measurement uncertainty of 0.40 × 10

^{−6}m (

*k*= 2).

## 6. Long distance measurements

To study dynamic FSI measurements of distances up to 20 m an experiment was set up to measure a range of distances with the dynamic FSI system in conjunction with a laser tracker.

#### 6.1. Experimental set-up

An experiment was set-up in which an SMR could be moved through a range of distances (0.2 m to 20 m) while being measured by the dynamic FSI system. Measurements were performed close to the floor in a section of corridor supplied with temperature controlled air in which there were no heat sources other than overhead lighting. The specification for the air is 0.5 °C and the temperature was measured with a calibrated sensor, accurate to better than 0.1 °C. The temperature was observed to be stable over the time of the experiment and an uncertainty of 0.5 °C is assumed.

The SMR was also monitored simultaneously by an API tracker (API tracker model three). The API tracker was set-up so that its output beam was directed over the top of the FSI launch collimator, as shown in Fig. 18, this is so that both beams travel though approximately the same air, and that any motion of the target is seen equally by both the API tracker and the dynamic FSI system. The API laser tracker was operated in interferometer (IFM) mode, making displacement measurements using the He-Ne fringe counting interferometer. This experimental set-up does not use a LaserTRACER and dichroic beam splitter as described in section 3 because the LaserTRACER is not designed to function through the dichroic beam splitter when there is a large OPD. The API tracker can measure 3D co-ordinates (*R*, *θ*, *ϕ*) and so its beam is not required to be exactly co-linear with the FSI laser and therefore the dichroic beam splitter is not required.

The target was moved to a range of positions and at each position thirty dynamic FSI scans and thirty sets of two hundred API tracker measurements were performed. The API tracker recorded the air temperature, pressure and humidity. API tracker and FSI measurements were corrected for the refractive index of air using these values. The dynamic FSI measurements used the 165 mm hydrogen cyanide cell calibrated as described in section 3 and the FSI lasers used the parameters shown in table 6. The laser tracker used its conventional interferometer, with a stabilised HeNe laser, and angle encoders to measure the positions of the target.

#### 6.2. Analysis of long distance measurements

The API tracker provides measurement points in spherical co-ordinates (*R*, *θ*, *ϕ*) with the center of the co-ordinate system being the intersection of the API tracker’s angular motion axes. As the dynamic FSI system only measures distances, the measurements from both systems were compared using a mathematical least squares model. In the model *D _{j}* is the distance measured by FSI, (

*R*,

_{j}*θ*,

_{j}*ϕ*) are the co-ordinates measured by the API tracker, (

_{j}*x*,

_{FSI}*y*,

_{FSI}*z*) are the Cartesian co-ordinates of the FSI launch collimator,

_{FSI}*S*is the scale factor between the two measurement systems and

*j*is the measurement number which ranges from 1 to

*n*. The model equation

*F*(

*X*,

*L*) = 0 is:

*X*) is:

*L*) is:

*n*respectively as defined below:

*P*is the 4

*n*by 4

*n*weight matrix.

*P*in this case is diagonal with the elements being the inverse measurement uncertainties squared.

*W*=

*F*(

*X*,

*L*) for the current

*X*values and

*X*

_{0}is the a starting estimate for

*X*. Equation (26) is solved iteratively until

*F*(

*X*,

*L*)

^{2}is minimised. The uncertainties of the dynamic FSI measurements are described in section 5 and the relative uncertainty for the API tracker measurements for

*R*is 0.5 × 10

^{−6}(

*k*= 2), for

*θ*is 0.35 × 10

^{−6}(

*k*= 2) and for

*ϕ*is 0.35 × 10

^{−6}(

*k*= 2).

#### 6.3. Results of long distance measurements

The model described in section 6.2 is solved and the fit residual (final values of the left hand side of Eq. (22) are shown in the upper plot of Fig. 19. The residual is divided by the FSI distance to get the relative residual and is shown in the lower plot of Fig. 19. The relative residual is approximately constant over the range of distances and a histogram of the relative residual is shown in Fig. 20.

The upper plot in Fig. 19 shows a residual which is symmetrical around zero, without a systematic trend, showing that the experiment and the fit are consistent. The Gaussian fit to the histogram of the relative residual, shown in Fig. 20 gives a one sigma width of 0.124 × 10^{−6}, which at *k* = 2 is 0.25 × 10^{−6}, is slightly better but still in agreement with the statistical spread observed during the calibration experiments described in section 3. The value of the parameter *S* given by the fit with 1.0 subtracted is 0.090 × 10^{−6} ± 0.19 × 10^{−6} (*k* = 2) and therefore compatible with 0 and shows that the calibration of the gas cell has worked.

## 7. Conclusion

By introducing a gas cell into a dynamic FSI system, the dynamic FSI equations can be solved without the use of non-linear least squares and the length of the reference interferometer can be determined for each scan. The ability to determine the reference length at measurement time removes the requirement for the reference interferometer to have a calibrated length which is stable over very long time intervals. The removal of this requirement allows the use of more compact and inexpensive interferometers to replace the complex and expensive Invar interferometers. The highly stable gas cell is now the length standard for the system and is traceable to international standards through a calibration procedure. Two different length gas cells were calibrated and it was seen that the longer gas cell gives the better results. The longer gas cell gives the better statistical uncertainty because it has deeper absorption features which allow more precise fitting. A dynamic FSI system using the calibrated longer gas cell can measure absolute distances of up to 20 m with relative measurement uncertainties of 0.40 × 10^{−6} (*k* = 2) with a sub scan resolution of 40 nm. This statistical uncertainty is compatible with the expected uncertainty of measurements though air, due to turbulence.

The dynamic FSI system can measure lengths at the sampling rate of the data acquisition electronics, in this case 2.77 MHz, which allows highly time resolved measurements of variable lengths. The combined absolute and sub scan capabilities allow the measurement of highly time resolved absolute distances without any prior knowledge of that distance.

The dynamic FSI system described here has the capability to measure the length of many measurement interferometers simultaneously. The maximum number of measurement interferometers is limited by the laser power and the ability to acquire their data and analyse that data in a reasonable time. The amount of laser power can be easily and cost efficiently increased with the use of commercially available EDFAs, leaving the number of measurement interferometers limited only by the data acquisition system. The performance of the system is good enough to be limited by air turbulence. A commercial version of the dynamic FSI system described here is now available from Etalon AG [23].

## Acknowledgments

We would like to acknowledge the funding support from EPSRC (grant EP/H018220/1), STFC ( ST/I000526/1), National Physical Laboratory and the University Of Oxford department of Physics. NPL acknowledges support from the BIS NMO National Measurement System Programme for Engineering and Flow Metrology. We would also like to acknowledge the in kind support from Etalon AG and in particular its CEO Dr. Heinrich Schwenke.

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