Absolute distance measurements for in-situ interferometer characterisation using range-resolved interferometry

The first method to be used for absolute distance measurement is the direct evaluation of the range-dependent demodulation phase carrier amplitude $A_\mathrm{d}$ [33] in RRI. This is analogous to the tomographic view as a function of the OPD provided by a Fourier transform of the interferogram in swept-source frequency domain optical coherence tomography [35], frequency scanning interferometry [25], optical frequency domain reflectometry [36] or FMCW lidar [37]. These techniques use linear (sawtooth ramp or triangular) laser frequency modulation waveforms and, in ideal conditions, a simple Fourier transform of the resultant interferogram provides a tomographic view of the signal amplitude as a function of OPD of the constituent interferometers present. In contrast, RRI uses low-complexity sinusoidal laser frequency modulation waveforms where only a sinusoid constituting a single frequency component in applied, but where consequently a more complex demodulation approach using a series of unique, range-dependent carrier signals is required to obtain the tomographic information. In general, the complexity with linear techniques arises due to the practical difficulty of obtaining linear waveform sweeps because of the many harmonics involved in the modulation waveform [38, 39]. Therefore linear techniques require either complex laser designs that can natively perform linear sweep waveforms, the use of a sweep-reference interferometer with associated signal processing chain to re-sample the data acquisition to match the actual sweep conditions or the adaptive pre-shaping of the sweep waveform [39]. RRI on the other hand, due to its simple and consequently very stable modulation regime can work in a very basic setup without sweep-reference interferometer, using only a monolithic laser diode, thereby moving complexity from the optical to the electronic signal processing domain, where it is likely to lead to more cost-effective and robust solutions.

In the setup for the present paper, the contributions of two signal sources at different ranges, the reference and the target interferometers, are simultaneously present in the interferogram. By comparing the ratio of two range values, and with the reference range at a known length, the absolute target distance can be determined. The principle of the method is illustrated in figure 1. Here, the interferogram is evaluated and the demodulated signal amplitude plotted against the demodulation phase carrier amplitude $A_\mathrm{d}$ [33], i.e. the phase excursion of the carrier that is used to demodulated the interferogram resulting from sinusoidal laser frequency modulation in RRI. Using the assumption that the measured distances are short compared to the distance corresponding to the time-of-flight of the sinusoidal modulation waveform of period $T_\mathrm{mod}$, the $A_\mathrm{d}$ value of the peak position should be directly proportional to the OPD of an interferometer present in the setup [33] and an example tomographic view is shown in figure 1(a). The determination of the peak positions then allows the OPDs of each signal source present to be determined.

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In the proposed technique, in order to make it compatible with real-time measurements, the signal amplitude is evaluated at a limited number of regularly spaced evaluation locations along the $A_\mathrm{d}$ axis, termed range channels. Depending on the chosen Gaussian window width parameter used in the RRI demodulation [33], where a Gaussian window width parameter of σ = 0.05 is used in this work, a distance between range channels of $1~\mathrm{rad}$ is chosen. However, it is stressed that this is only an initial choice and a detailed investigation on the optimal parameter choices is still outstanding. In this way, range channels are close enough that several evaluation locations (at least 3) cover the peak in every possible position, as illustrated in figure 1(a). Because the shape of the peak is defined by the Gaussian window function used in the RRI signal processing it will also be a Gaussian shaped exponential function of the form $f(x) = \exp{(-x^2)}$ in the $A_\mathrm{d}$ plot. Consequently, the natural logarithm of the range signal will correspond to a parabola, as shown in figure 1(b). When the local slope of the parabolic signal at the evaluation locations is calculated through differentiation, a straight line will then result as shown in figure 1(c). The zero-crossing of this line equals the desired peak position, and is found through linear interpolation from the two range channels just before and just after the sign change. Please note that in order to preserve symmetry, the value of the differentiation of a pair of range channels is assigned to the middle of the range in between them, as also illustrated in figure 1(c).

A further method for absolute distance measurements is the use of an additional, slow laser frequency modulation waveform, leading to a range-proportional phase signal after phase demodulation for each constituent interferometer. The amplitudes of these phase signals can then be recovered using lock-in techniques. To implement this method, the injection current of the laser diode is modulated simultaneously with two sinusoids, one fast, high-amplitude sinusoid with modulation period $T_\mathrm{mod}$, identical to the one used in the first method in section 2.1, and one additional slow, low-amplitude sinusoid of period $T_\mathrm{mod,lock-in}$. Here, the fast modulation signal is used to perform interferometric signal processing and phase determination using the RRI technique in the usual way [33], while the slow modulation then yields a measurable phase signal whose amplitude can be determined. As in all laser wavelength modulated interferometers, the amplitude of the phase excursion is directly proportional to the OPD of the constituent interferometer.

In the proposed method, the phase of the interferometer is demodulated at a single range channel each for both reference and target interferometers, approximately located on the top of the respective peaks as illustrated in figure 2(a). The resulting phase signals over one modulation period $T_\mathrm{mod,lock-in}$ of the slow laser frequency modulation waveform are shown in figure 2(b). The phase excursion amplitude recorded for each constituent interferometer should then be directly proportional to its OPD and the amplitude of the sinusoids can be easily recovered through lock-in demodulation as the applied modulation is synchronous to the data acquisition. Again, by calculating the ratio between the phase excursion amplitude of the target and the reference interferometers, the target distance can be determined.

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An example photo detector signal corresponding to a single interferogram over one modulation period $T_\mathrm{mod}$ is plotted in figure 5(a), along with the Gaussian window function with window width parameter σ = 0.05 used in the RRI signal processing [33] to select the parts of the modulation waveform with the highest fringe frequencies. The resulting range dependency is evaluated from the interferogram as a function of demodulation phase carrier amplitude $A_\mathrm{d}$ and is shown in figure 5(b). Due to the non-linearity of the laser frequency modulation that occurs in every laser diode even when a perfectly sinusoidal injection current modulation waveform is applied, higher order non-linear corrections of the modulation waveform need to be incorporated into the demodulation carriers. Here, a first carrier harmonic of 4.5% at a phase shift of 5^∘, as determined by visual inspection of the symmetry of the demodulation phase carrier amplitude versus signal processing delay maps, was used in the demodulation carriers in addition to the correction of parasitic intensity modulation as previously described [33].

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In the resulting tomographic view shown in figure 5(b), the reference peak has its maximum at $40.35~\mathrm{rad}$ and the target peak has its maximum located at $106.32~\mathrm{rad}$, with full-width half-maximum (FWHM) peak widths of $8~\mathrm{rad}$. The physical path length of the reference peak is known through the spacing of the gauge block to be $50.000~\mathrm{mm}$. Therefore, the scaling factor is $1.239~\mathrm{\frac{mm}{rad}}$ and the physical path length of the target peak can be determined as $131.75~\mathrm{mm}$, with a corresponding peak width (FWHM) of $10~\mathrm{mm}$. A further peak with low amplitude can be seen at ${\approx}80~\mathrm{rad}$, which is likely to be caused by a undesired multiple reflection (double bounce) signal in the reference interferometer. Therefore, figure 5(b) also provides an example of how the tomographic view can be used for in-situ analysis of an interferometer and the detection of multiple reflection signals which are physically present within the interferometer assembly.

In this configuration, in order to ensure that the double bounce signal does not overlap with a target signal, the target signals are restricted to demodulation phase carrier amplitudes well above the double bounce peak. Thus the stage is positioned so that the stage travel range is located between $A_\mathrm{d} \approx 94~\mathrm{rad}$ and $A_\mathrm{d} \approx 135~\mathrm{rad}$, i.e. between 117 and $167~\mathrm{mm}$. However, as also discussed later, if two separate photo detectors, ADCs and associated fibre components would be used to individually evaluate the target and reference interferometers, no such restrictions would apply and a much wider travel range could be covered.

To implement the lock-in technique of section 2.2 the target and reference interferometers are simultaneously demodulated and the amplitude of their respective phase response at the additional modulation frequency $1/T_\mathrm{mod,lock-in} = 20~\mathrm{Hz}$ is analysed. Measured phase signals over one lock-in period are shown in figure 6(a) with peak-to-peak phase amplitudes of 22.09 and $58.20~\mathrm{rad}$ for the reference and target interferometer, respectively. The phase modulation depth of the lock-in signal was chosen to be close to the maximum fringe rate permissible (${\approx}\frac{0.5}{T_\mathrm{mod}}$) at the maximum target range to make use of the highest possible phase signal excursions. Because, as discussed previously, the injection current modulation of a laser diode generally results in a non-linear optical frequency modulation, a first harmonic of the lock-in frequency at non-zero amplitude exists, as visible in figure 6(b), however, due to the lock-in principle this is of no concern to the amplitude measurement of the fundamental sinusoid. Figure 6(c) then plots the residuals of the phase signals after the fundamental sine and the first harmonic are subtracted. The residual signals exhibit standard deviations of 0.04 and $0.06~\mathrm{rad}$ for the reference and target interferometers, respectively, and this will be, in general, caused by a combination of noise and cyclic errors. However, because no characteristic fringe-speed dependent signal patterns with π or 2π phase distance that would be indicative of strong cyclic errors are discernible and also because very low non-linearities at the mrad levels have previously been demonstrated [33, 43] with RRI, is is likely that the bulk of the residual signals is actually due to noise and not due to cyclic errors. This is important because, unlike noise, cyclic errors could lead to systematic errors in the distance determination for the lock-in technique and therefore they need to be sufficiently low.

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First, in order to investigate the general quality of the measurement data for the two proposed methods, results from an experiment using a $0.5~\mathrm{mm}$ ramp movement are shown in figure 7, recorded over a duration of $10~\mathrm{s}$ and using the external interferometer to verify the measurements. Here, on the left panels, in figures 7(a) and (b), the results from the direct peak detection method of section 2.1 are shown, while the plots on the right, in figures 7(c) and (d), show results from the lock-in method of section 2.2. Note that these results were recorded at separate instances for the two methods, thus the motion pattern is not exactly identical. In order to allow better comparison of the noise performance of the two methods, the data rate should be identical for both cases. Therefore the results for the direct peak evaluation method are averaged over $1/20~\mathrm{Hz}$, identical to the native data rate of the lock-in method $1/T_\mathrm{mod,lock-in}$.

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Figure 7(a) shows the raw peak position measurements for the reference and target interferometers, respectively, while figure 7(b) shows the calculated target distance measurement along with the external verification interferometer displacement measurement fitted to the initial, stationary part of the absolute distance measurement. The ramp measurement was repeated using the lock-in method, with results plotted in a similar manner in figures 7(c) and (d). Comparing the distance results for the two methods, it can be seen that the noise levels for the direct peak evaluation method are considerably lower than for the lock-in method. In the static parts of the signals noise standard deviations (1σ) are 18 and 76 μm for figures 7(b) and (d), at $20~\mathrm{Hz}$ data rate, respectively. A key difference between the two methods lies in the ability to measure during target movements, where the direct peak evaluation method can correctly measure the position during the entire ramp movement, as seen in figure 7(b), while the lock-in method completely fails during the movement of the stage, as seen in figure 7(d). The reason is thought to be due to frequency components in the demodulated phase signal of the ramp movement coinciding with the lock-in frequency and distorting the measurement. In future implementations, this behaviour could be reduced by using higher lock-in frequencies further away from possible target phase signal frequency content, however, in the present setup using the low-cost Nucleo-board, the lock-in method is clearly only suitable for static measurements.

To estimate how linear and accurate both methods perform over a larger distances, the following measurement campaign is performed: the linear stage shifts the target in $5~\mathrm{mm}$ steps over a total distance of $50~\mathrm{mm}$, where at each step 10 measurements are taken, each lasting $1~\mathrm{s}$, then the mean value and the standard deviation at each stage position, indicative of the repeatability of the measurement, is calculated. Calculated mean values of the target and reference peak position for the direct peak evaluation method and of the target and reference phase amplitude for the lock-in method are shown in figures 8(a) and (d), respectively. The measured absolute target distance at each stage position coincides well with the relative displacement data measured by the external interferometer as shown in figures 8(b) and (e). Considering the target distance value at the first stage position as an initial null displacement value allows to directly calculate the error of estimated distance measurements with respect to the external interferometer readings.

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As plotted in figures 8(c) and (f), noise standard deviations over ten $1~\mathrm{s}$ measurements at a single location vary from 2 to 13 μm for the direct peak evaluation and between 9 and 42 μm for the lock-in method. Therefore the noise standard deviations in the results for the lock-in method exhibit a significantly larger variation at different stage positions in comparison to the direct peak evaluation method, but the direct peak evaluation method also shows some variation. The reason for this wide variation is not immediately clear, but could be explained by different noise levels for different range channels, varying signal strengths or perhaps changes in environmental vibration levels at different stage positions.

Crucially, in figures 8(c) and (f) the mean values of the resulting cumulative distance errors are within ${\pm40}~\unicode{x03BC}$m for both the direct peak evaluation method and for the lock-in method, equating to fractional errors below ±0.1% over the $50~\mathrm{mm}$ travel range. This experiment therefore provides a first estimate of the achievable accuracies for both methods and the accuracy levels demonstrated here are deemed to be sufficient for the purpose of interferometer characterisation and dead path determination in most cases. A separate setup using two adjacent, gauge-block spaced interferometers with transparent windows and illuminated by a single beam [44], were gauge blocks in variable combinations could be inserted, could allow a more formal quantification of the uncertainties involved. This experiment would directly provide an absolute independent reference measurement provided by the additional gauge block, rather than the relative independent reference measurement currently provided by the Helium-Neon interferometer, simplifying the uncertainty analysis, and would also be free from geometrical errors due to the sequential single-beam arrangement, however, such a dedicated setup is beyond the scope of this paper.

A long-term measurement is performed to determine the stability and the influence of drift of the laser modulation on the measured distance. This measurement lasted 8 h each for both methods, where the target distance was averaged over $1~\mathrm{s}$ of data with an interval between measurements of 19 s during which the signal is processed, leading to a sample rate of $\frac{1}{20~\mathrm{s}} = 0.05~\mathrm{Hz}$. The results are shown in figure 9, again for the direct peak evaluation method on the left panels and for the lock-in method on the right panels, where both experiments were conducted on different nights and are therefore not correlated. For all panels in figure 9, the secondary y-axis is labelled in fractional deviation from the mean in units of parts-per-million (ppm). Here, figures 9(a) and (b) as well as figures 9(d) and (e) show the time series of the reference and target signals, respectively, while the lower panels, figures 9(c) and (f), show the computed distance measurements.

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The obtained results demonstrate that drifts that affect the target signal and the reference signal proportionally, for example a drift in the laser modulation depth, cancel during the distance calculation. For example, for the direct peak evaluation method shown in figure 9 on the left panel, a small jump in the first hour of measurements is clearly visible, with proportionally the same magnitude for the reference peak position and the target peak position shown in figures 9(a) and (b), respectively. However, it can then be seen in figure 9(c) that this does not affect the final calculated distance measurement. As for the lock-in method, there is a clear upward drift for both the target phase amplitude and the reference phase amplitude shown in figures 9(d) and (e), which does not occur in the calculated distance shown in figure 9(f). The computed standard deviations for the $1~\mathrm{s}$ averaged data are 2.7 μm, or $23~\mathrm{ppm}$, for the direct peak evaluation method in figure 9(c) and 5.0 μm, or $43~\mathrm{ppm}$, for the lock-in method in figure 9(f).

To further investigate the stability behaviour of the techniques, the overlapping Allan deviation [45] of the computed distance measurement data of figures 9(c) and (f) was calculated and is shown in figure 10. Here it can be seen that for averaging times between $20~\mathrm{s}$ until ${\approx}800~\mathrm{s}$, the Allan deviations drop with a slope of $\sqrt{10}$ per decade, as would be expected for a measurement dominated by white noise. Above ${\approx}800~\mathrm{s}$, the stability decreases again with the onset of drifts, where it is unclear whether these drifts are caused by the measurement device or by environmentally introduced thermal variations in the mechanical setup. However, at the optimum averaging period of ${\approx}800~\mathrm{s}$, Allan deviations below $10~\mathrm{ppm}$ were recorded for both techniques, equating to noise standard deviations below 1 μm over a typical measurement distance of $0.1~\mathrm{m}$.

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A tomographic view of the interferometric assembly can be very useful to qualitatively identify multiple reflections or other parasitic signal sources, both by locating the OPDs of the error sources and also by measuring the relative signal strengths. In RRI this functionality is provided by the plot of the signal amplitude as a function of demodulation phase carrier amplitude $A_\mathrm{d}$, as shown in figure 5(b). The minimum separation between any two signal sources, i.e. the two-point resolution, is given by the FWHM widths of the peaks in figure 5(b). In this implementation, this value was limited to $10~\mathrm{mm}$ due to hardware limitations on the available modulation signal amplitude, leading to a scale factor of $1.239~\mathrm{\frac{mm}{rad}}$. However, it is expected that even for the laser diode currently used, increasing the injection current modulation amplitude would improve this value easily by more than a factor of two to below $5~\mathrm{mm}$ (in air), in line, or even better, than the scale factor of $0.58~\mathrm{\frac{mm}{rad}}$ that has previously been achieved [41].

Also, it has already been mentioned in section 3.1 that for the setup presented in this paper, if two independent photo detectors would be used to interrogate the target and reference interferometers separately, then no restrictions on the target travel range due to multiple reflections of the reference interferometer would apply. However, it should be noted that the proposed technique cannot be used for measurements around the zero-OPD balance point but that otherwise there is no principal restriction of the approach to work over much larger travel ranges than the 50 mm demonstrated here. Any signal source that can be resolved must have an OPD larger that the two-point resolution of the system. In general, to simplify the setup in situations where only a course measurement is required, for example to qualitatively evaluate the tomographic view, the continuous interrogation of the reference interferometer as used in this work may be omitted and the distance scaling factor can be calibrated before the measurement, for example by the temporary use of a gauge block-spaced reference interferometer. It can be seen from the raw measurement data in figures 9(a) and (b) that the modulation parameters can be expected to remain stable to levels on the order of several $100~\mathrm{ppm}$ over several hours. Therefore it can be reasonably expected that even without continuous online calibration as used in this paper, uncertainties in the scaling factor should still remain well below $1000~\mathrm{ppm}$ or $0.1\%$. Note that this would constitute uncertainty that is in addition to any position-dependent systematic errors that were established in figures 8(c) and (f).

In addition to providing the tomographic view of the interferometer assembly, the proposed approach also allows for the quantitative measurement of the absolute distances of the constituent interferometers using the two methods described in section 2. It was demonstrated in figure 8 through comparison with an external verification interferometer that cumulative systematic errors for both methods are below ${\pm40}~\unicode{x03BC}$m over the $50~\mathrm{mm}$ travel range. If laser diodes at the appropriate wavelength for the targeted interferometer assembly are used, for example at $633~\mathrm{nm}$ or $780~\mathrm{nm}$, the methods described in this paper could be readily applicable to many types of commonly-used interferometers, including Michelson and Fabry–Perot configurations, with both fibre-coupled or free-space interfaces, where especially fibre-coupled interferometers [32, 41, 46] often work in tightly enclosed spaces and use compact optics, and where external geometric measurements, for example of dead paths, are therefore often not feasible.

The proposed approach could be used during initial setup, temporally replacing the main measurement laser of the interferometer to be diagnosed. Alternatively, the approach could even be employed for continuous operation alongside a regular precision displacement measuring interferometer, for example alongside a Helium-Neon laser interferometer, using a laser diode with a wavelength that is slightly different from the main interferometer and using filters to separate the respective signals.

4.4.1. Improvements to the direct peak evaluation method.

4.4.2. Improvements to the lock-in method.