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Abstract
The change in length of an optical fiber-based Fabry-Pérot cavity (FPC) can be precisely measured using phase tracking, but the displacement range is limited by phase ambiguity. Period tracking techniques determine the absolute FPC length, but with larger uncertainties from tracking the spacing between multiple peaks. A hybrid method is demonstrated that identifies appropriate peaks for phase tracking using a coarse estimate obtained from the free spectral range to effectively maintain the high precision (∼1 nm) of phase tracking techniques to measure ∼24 µm displacements, well beyond the range limitations (typically <1 µm) of phase tracking methods.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical fiber-based sensors are routinely used for extremely high-precision measurements of various parameters, including length, displacement [1], and various engineering process parameters that can be derived from changes in length, such as temperature [2], pressure [3], fluid flow rate [4,5], liquid level [6,7], and vibration [8]. Their immunity to electromagnetic interference, small footprint, remote interrogation, and ability to survive harsh environmental conditions also make them candidates for process monitoring in industrial applications, such as chemical processing or energy production applications [9,10]. For example, fused silica [11–13] and single-crystal sapphire [14,15] optical fibers transmit light at temperatures approaching 1,000 and 1,400$^{\circ }$C, respectively. Their ability to survive relatively high doses of neutron [16–21] and gamma [22–24] radiation also makes them attractive candidates for nuclear reactor applications.
Recent work has focused on leveraging the high-precision displacement monitoring capabilities of fiber-optic sensors to develop a sensor to measure corrosion in nuclear reactors and in experiments performed in research reactors [25]. The sensor is fabricated using ultrasonic additive manufacturing (UAM) [26,27] to directly embed an optical fiber within a metal matrix [28–31]. The surrounding metal fixes the terminal end of the fiber within a metal housing. The metal housing terminates with a thin metal diaphragm that deflects when subjected to a differential pressure [32]. In this way, a Fabry-Pérot cavity (FPC) is established between the terminal end of the embedded optical fiber and at the inner surface of the metal diaphragm. This approach protects the optical fiber from the surrounding corrosive fluid and enables the sensor to be actuated via internal pneumatic pressurization. Corrosion can be monitored by calculating the change in the diaphragm’s thickness based on the analytical expression that relates the measured diaphragm displacement to the applied internal pressure [25]. With an appropriate interrogation technique, the same sensor could also measure higher frequency dynamic fluid interactions such as flow-induced turbulence or vibration.
The ability to accurately resolve micrometer-scale changes in thickness requires measuring diaphragm displacements on the order of micrometers to tens of micrometers with an accuracy on the order of tens of nanometers. Prior works [33,34] have applied spectral transforms (fast Fourier transforms and cepstral analyses), up-sampling, and curve fitting to the measured interference spectra in an effort to achieve nanometer-scale precision. These spectral techniques come at the cost of increased processing complexity. Wavelet transform-based techniques have also been utilized to demodulate change in FPC length. For example, a Fourier transform was used to generate a plausible range of FPC deflections, which was searched using wavelet transforms to identify the FPC deflection with greater precision [35,36]. Similar to other spectral transforms, the complexity of the wavelet transform analysis makes it challenging to implement and quickly process data, particularly for high frequency applications.
More recently, a custom-designed low-coherence interferometer (LCI) was developed and used novel phase-demodulation scheme to resolve changes in FPC length [32,37,38]. Although this system achieved the required resolution, the range of displacements that could be measured were limited to $\pm$ $\sim$15 $\mu$m based on the coherence length of the light source. Moreover, because this system measured total displacement by tracking the cumulative sum of incremental changes in FPC length (i.e., phase tracking), it was susceptible to errors that could be introduced during rapid changes in the FPC length caused by changes in temperature or differential pressure, which could propagate through future measurements [39]. LCI systems can also suffer from signal attenuation, even with compensation techniques [40,41].
This motivated interest in developing a simple method for monitoring larger FPC length changes with a higher dynamic range (e.g., systems that use tunable lasers) and a resolution on the order of tens of nanometers or better. Such a method would have a much broader applicability to any FPC-based sensor that would benefit from the high precision of phase tracking methods over a wider measurement range. Optical coherence tomography techniques have been developed using spectral multiplexing [42] and phase gradients [43] to introduce secondary features into the interference pattern to extend the dynamic range of the measurements while maintaining the precision of phase tracking. Other interrogation methods have achieved similar results with free-space optics by using the optical frequency comb of a femtosecond laser pulse [44] and a pseudo-dithering scheme based on laser feedback [45]. Because the ultimate application for the present work is for sensors deployed in harsh environments, free-space optics would be challenging to implement. Therefore, this work focuses on optical fiber-guided signals.
2. Methods
2.1 Measurement theory
An FPC is an optical resonator that comprises of two parallel reflective surfaces separated by a medium with group index $n_g$ and cavity length $l$. Light that is incident on the first surface—usually launched via an optical fiber—is partially reflected. The transmitted portion of the light passes through the medium and is reflected at the second surface before it recombines with the light that was reflected from the first surface to generate an interference pattern. The intensity of the reflected interference pattern (reflectance, $I_R$) generated by an ideal lossless FPC is given by
Relative displacements can also be measured from the same spectral interference pattern using phase tracking methods. With this approach, one or multiple peak wavelengths in the interference pattern are tracked over time, and the relative change in cavity length is proportional to the relative change in the peak wavelengths. Equation (4) describes the relationship between the relative change in cavity length ($\Delta l_m(t) = l_m(t) - l_m(t_0)$) and the change in wavelength $[\Delta \lambda ]_m$ for peak $m$ relative to the initial wavelength at time $t_0$:
Because phase tracking methods depend only on the change in one peak, they can resolve much smaller displacements [48]—nanometer resolution is possible—but the maximum resolvable change in cavity length is limited because of phase ambiguities. Per the definition of $\phi$, the maximum resolvable change in cavity length for an air cavity ($n_g=1$) is equal to $\lambda /2$, or less than 1 $\mu$m for typical wavelengths on the order of 1,550 nm. This range limitation can be overcome by performing a cumulative sum of incremental phase shifts, but aliasing errors can still occur if the incremental phase shift exceeds 2$\pi$ [32].2.2 Approach
Drawing inspiration from [35], a predictor-corrector scheme is described herein to realize the high precision of phase tracking techniques while also measuring absolute deflections greater than the limitations imposed by the FSR (i.e., phase ambiguity). Figure 1 illustrates the scheme with an example interference pattern, and Fig. 2 provides a flow diagram of the proposed scheme. First, Eq. (3) is used to provide a coarse measurement of the initial cavity length, $l_m(t_0)$, and peak locations $[\lambda ]_m(t_0)$. A mean operation, including a removal of the upper and lower 25% of the length estimates $l_m(t_0)$ to mitigate the effect of outliers, is used to determine $l(t_0)$ from $l_m(t_0)$. This process is then repeated for all $t$ to determine the time-dependent cavity lengths $l(t)$. The predicted peak wavelengths ($[\lambda _P]_m(t)$) for $t>0$ are calculated using the change in $l$ and the initial peak locations ($[\lambda ]_m(t_0)$) as
From this coarse estimate, the corrected peak locations ($[\lambda _C]_m(t)$) are identified by searching for the closest peaks to $[\lambda _P]_m(t)$. The corrected peak locations are then used to precisely determine the change in cavity length $\Delta l_m(t)$ (Eq. [6]), or $\Delta l(t)$ after taking the median over all $m$ for which $\lambda _{min}<[\lambda _P]_m<\lambda _{max}$:
Fig. 1. Schematic of (a) an FPC and (b) the associated interference patterns showing the reflected intensity $I_R$ from an idealized FPC and the peaks shifting from blue to red as the cavity length changes. The true wavelength shift $\Delta \lambda _m$ is shown, along with the coarse estimate $\lambda _P$. The upper plot shows the probability density $\rho$ for the predicted wavelength $\lambda _P$ determined using period tracking. The peak nearest $\lambda _P$ is then identified as $\lambda _C$.
Fig. 2. Schematic flow diagram for the hybrid phase-period demodulation scheme where $k$ is the sample window used for the temporal median filter.
2.3 Hardware and data acquisition
The predictor-corrector scheme was evaluated using a sensor that was described previously for measuring the external pressure [32] or corrosion [25] of a thin metal diaphragm. The FPC was formed by first by using UAM to embed an optical fiber in a nickel-200 block, which was then welded within a cylindrical nickel-200 housing. The housing was terminated with a mirror-polished thin diaphragm with the fiber terminus separated from the diaphragm’s surface by $404 \pm 5$ $\mu$m (measured using period tracking). Even after polishing, a thin film could remain on the inner surface of the diaphragm, which could lead to the creation of a dual-cavity FPC, as described in [49,50]. However, no such artifacts were observed during previous testing of similar sensors [25,32]. Diaphragm deflections were initiated either by manually depressing the outer surface of the diaphragm or via internal pneumatic pressurization of the sensor. The FPC was interrogated using a Hyperion si155 optical sensing instrument (Luna Innovations Inc., Blacksburg, Virginia, USA). The sensing instrument contains a TLS with a wavelength range of 1,460 to 1,620 nm (Fig. 3). A photodetector (PD1) records light launched by the TLS, which passes through a reference fiber located on the lower-amplitude arm of a 90/10 coupler. The other arm guides the light from the TLS into the FPC through a 50/50 coupler. The reflected light from the FPC passes through the other input arm of the 50/50 coupler and is recorded by a second photodetector (PD2). The optical spectra recorded by PD1 and PD2 are combined to produce the interference pattern resulting from light reflected from the FPC. The optical interrogator includes a peak-finding algorithm that returns the peak locations at a sampling rate of 1 kHz. These peaks were first analyzed using phase tracking, period tracking, and hybrid algorithms implemented in Python 3.8.11. These same algorithms were later integrated into the data acquisition software in LabVIEW 2019 (version 19.0f2; National Instruments, Austin, Texas, USA).
Fig. 3. Schematic showing the optical network in the Hyperion si155 optical interrogator used to measure the FPC length.
3. Results
3.1 Simulated deflections
Although the proposed demodulation scheme is not subject to phase ambiguity like phase tracking techniques, it is limited by the range of the TLS. Rearranging Eq. (4) such that $[\lambda ]_m(t)$ spans the TLS range $[\lambda _{min}, \lambda _{max}]$, the FPC length at time $t$, relative to the initial length, is bounded by
To characterize the theoretical performance of the proposed hybrid demodulation scheme, FPC length changes were simulated over 99% of the allowable deflection range (Fig. 4). Figure 4(a) shows the imposed FPC deflections used to generate the simulated interference patterns and the FPC deflections that were determined using the hybrid scheme with the addition of random errors in the peak identification (wavelength domain). These random errors are meant to simulate the effects of false peak identification resulting from the coarse estimate obtained from period tracking. As the cavity length was varied vs. time (Fig. 4(a)), the added error was randomly sampled across a uniform distribution between $\pm 1$ nm ($<[\Delta \lambda _{fsr}]/2 \approx 1.5$ nm per Eq. (2)) and $\pm 3$ nm ($>[\Delta \lambda _{fsr}]/2$). Figure 4(b) shows the error between the imposed vs. demodulated changes in the FPC length as a function of time (left side) and as a histogram (right side). The apparent discrete nature of the error distribution results from false peak identification during the coarse estimate (predictor) step in the algorithm, which is evident when the peak wavelength error exceeds $[\Delta \lambda _{fsr}]/2$. This phenomenon is discussed in more detail in Section 3.4.
Fig. 4. Simulated FPC deflections demodulated using the proposed hybrid scheme. The calculated deflections with $\pm 1$ nm and $\pm 3$ nm error applied to the peak wavelength selection, along with the imposed deflections (a), and the difference between the calculated vs. imposed deflections are also shown (b), with a histogram with 40 nm bins showing the symmetric discrete error distribution that results from identifying false peaks in the demodulation scheme.
3.2 Repeated large negative deflections
To demonstrate the proposed hybrid technique and compare the measured displacements with those obtained using conventional period and phase tracking techniques, the diaphragm was repeatedly manually depressed to produce deflections on the order of 15–20 $\mu$m (Fig. 5). As expected, the period tracking and hybrid methods were able to resolve deflections beyond the limitations of the phase-tracking method. The displacements determined by using the hybrid method matched those determined by using the period tracking method but with a noticeably higher precision.
Fig. 5. Experimental measurements of FPC deflections demodulated using period tracking (gray), phase tracking (red), and the proposed hybrid technique (blue) as the diaphragm was manually depressed many times over the course of 10 s. The upper plot shows the phase tracking data with the initial phase measurement used as a reference without performing the cumulative summation. This illustrates how phase tracking is unsuitable for tracking such large FPC deflections. The cumulative phase tracking method is shown in the lower plot to illustrate how cumulative sums of incremental displacements determined using phase tracking can lead to error propagation.
3.3 Step-wise positive deflections
More controlled diaphragm deflections were initiated by pneumatically pressurizing the interior of the sensor. A split-rail proportional–integral–derivative (PID) controller, described previously [25], was used to produce a step-wise pressure waveform. An independent pressure transducer was used to record the internal pressure simultaneously with the FPC displacements. Both sets of measurements were recorded at a rate of 1 kHz as the pressure set point was increased from 100 to 150 kPa in steps of 12.5 kPa, each lasting 30 s (Fig. 6). These data were analyzed with and without using median time-domain filtering with a 25 sample window (i.e., affecting $k$ in the gray box in Fig. 2). Figure 6(a) demonstrates that both the period tracking and hybrid demodulation methods produce deflections that closely follow the increasing pressure steps at steady state, which was considered to be the last $\sim$3 s of each hold-step before the PID set point was changed. To characterize the variation of the data around each steady-state point, distributions were created for all data, both hybrid and period tracking methods, within each $\sim$3 s steady-state window after subtracting the corresponding median deflection obtained using the unfiltered hybrid data at each pressure set point. For both the hybrid and period tracking techniques, results are shown with and without a time-domain median filter with a 25 sample window. Without time-domain median filtering, the period tracking method yields a similar steady-state standard deviation of $\pm 0.07$ to $0.09$ $\mu$m, whereas the hybrid method produces a standard deviation of $\pm 0.002$ to $0.2$ $\mu$m (Fig. 6(b), Table 1). However, the primary source of error when using the hybrid method originates when $\lambda _C$ is incorrectly identified as a peak to the left or right ($\lambda _{m-1}$ or $\lambda _{m+1}$ in Fig. 1), resulting in a discrete phase shift error of $\pm 2\pi$, or a discrete error of $\pm \lambda /2$ in the FPC length. These errors collectively manifest as impulse or salt-and-pepper noise and are visible near $\Delta l - \bar {l} = -0.7 \mu m \approx -\lambda /2$ in Fig. 6(b). Therefore, this collection of incorrect peaks inflates the standard deviation associated with deflection measurements made with the hybrid method. Applying median time-domain filters (Fig. 6(c)) is a simple way to mitigate these errors [51]. The median operation, rather than the mean, is more appropriate for removing errors that occur when the incorrect peak selection is skewed to one side (Fig. 6(b)), especially because the error distribution is discrete and appears in $<10\%$ of the measurements. This filter improved the precision of the period tracking measurement by approximately two- to three-fold and had relatively little effect on the steady-state deflection measurements (Table 1). However, the same filter applied to deflection measurements produced using the hybrid method resulted in up to a hundred-fold increase in precision because it mitigated the effects of incorrect peak selection.
Fig. 6. The internal FPC pressure was increased between 100 and $\sim$150 kPa and held for $\sim$30 s at each step. (a) The pressure measurements (gray, right axis) are plotted along with the deflection measurements (left axis) determined using the unfiltered hybrid (transparent blue) and period tracking (transparent red) methods, as well as the filtered hybrid (opaque blue) and period tracking (opaque red) methods. Histograms with a bin size of 20 nm were generated from the steady-state measurements comprising the last $\sim$3 s of each $\sim$30 s hold period and were compared for the (b) unfiltered and (c) filtered hybrid and period tracking methods. The histogram data are represented as the percent of total counts in each bin. The error is defined as the difference between the demodulated FPC length change and the value obtained using the mean of the unfiltered hybrid data at each pressure set point.
Table 1. Steady-state internal FPC pressures and deflections measured using the period tracking and hybrid methods during the last 3 s of each 30 s constant-pressure period. Data shown as mean(median) $\pm$ standard deviation.
3.4 Propagation of uncertainty
The propagation of uncertainty for each FPC interrogation method was analyzed using the variance formula, as in [48]. The error in $\Delta l$ when using phase tracking ($[\sigma _{phase}]_m$) for peak $m$ is given by
Relative to the initial FPC length, this error is independent of time and only depends on the wavelength uncertainty $\sigma _\lambda$ relative to the initial peak wavelength $[\lambda ]_{m}(t_0)$. The time-dependent error in in $\Delta l$ when using period tracking ($[\sigma _{period}]_m(t)$) is given byThe proposed hybrid method is essentially identical to phase tracking as long as the coarse estimate obtained using period tracking (Eq. (5)) properly identifies the true peak. If $[\lambda _P]_m(t)$ provides a sufficiently good approximation for $[\lambda ]_m(t)$ (i.e., within $-[\Delta \lambda _{fsr}]_m(t)/2 < [\lambda _C]_m(t) - [\lambda ]_m(t) < [\Delta \lambda _{fsr}]_m(t)/2$), then the correct corresponding peak is selected so that $[\lambda _C]_m(t) = [\lambda ]_m(t)$. However, if $[\lambda _P]_m(t)$ is outside this window, an incorrect peak will be selected for $[\lambda _C]_m(t)$, which differs from $[\lambda ]_m(t)$ by approximately an integer multiple of $[\lambda _{fsr}]_m(t)$ in addition to the small correction required to identify $[\lambda ]_m(t)$. Therefore, the error in the hybrid method $[\sigma _{hybrid}]_m$ can be represented as
The data in Table 1 demonstrates that standard deviation of the filtered measurements made using the hybrid method are on the order of 1 nm. If the distribution of $k_m$ is truly symmetric around $k_m = 0$ such that $[\sigma _{hybrid}]_m = [\sigma _{phase}]_m \approx 1$ nm, then Eq. (3) can be used with $l(t_0)=404$ $\mu$m and $[\lambda ]_{m}(t_0) \approx 1540$ nm to give $\sigma _\lambda = 4$ pm. The manufacturer of the optical interrogator quotes a wavelength accuracy and repeatability of 1 pm, which is a similar order of magnitude. This suggests that the error for the hybrid does indeed approach that of phase tracking.
4. Discussion
The hybrid method proposed herein for demodulating FPC lengths from their optical spectra demonstrably improves precision compared with period tracking methods without relying on cumulative sums of incremental phase tracking measurements, which suffer from error propagation and drift. Rather than relying on introducing secondary features into the interference pattern by manipulating the incident light or affecting the optical signal returning to the interferometer, as in previous work [32,42–45], the hybrid method demonstrated in this work uses the primary interference pattern from the FPC alone via a filtering scheme. This is particularly attractive because it enables the hybrid method to be implemented as a simple drop-in replacement for demodulating data from FPC-based sensors. Although the hybrid method overcomes the challenges associated with conventional peak-tracking algorithms, it is still limited by the wavelength range of the tunable laser used to interrogate the FPC. Ideally, the maximum wavelength shift observable using the hybrid method is equivalent to the tuning range of the TLS, although in practice this is more limited. In the present study, >50 identifiable peaks initially appeared in the interference pattern recorded by the optical interrogation system. As the FPC length changed, many of those initial peaks were shifted outside the range of the TLS. When the number of remaining peaks that fell within the range of the TLS decreased below approximately 10–15, the algorithm failed to reliably demodulate the FPC length. This limitation in the maximum resolvable FPC displacement could be overcome by updating the reference spectrum at intermediate steps similar to the adaptive methods developed for optical frequency domain reflectometry [52,53].
The application of a time-domain median filter to the FPC deflection measurements produced using the hybrid method effectively removes the impulse noise from the measurement but also could introduce a delay and smooths over short-duration physical FPC length changes. The effect of this filtering depends on the size of the sample window. Depending on the desired accuracy and measurement frequencies of interest, the time-domain filtering could be adjusted, as needed. For the original application mentioned herein (i.e., corrosion monitoring via step-wise, internal static pressurization of an FPC), the frequency limitations imposed by the time-domain filtering would not affect the ability to perform these measurements.
5. Conclusion
Optical fiber-based FPCs are a popular sensor for measuring fine displacements because of their high precision. Spectral interrogation methods for FPCs are particularly popular because they produce measurements with precision on the order of nanometers using phase tracking methods. However, the maximum resolvable displacements using methods are limited by phase ambiguities. Period tracking methods overcome this limitation and enable the calculation of an absolute FPC length but do not offer the same precision. To overcome the range limitation of phase tracking methods and the relatively poor precision of period tracking methods, a hybrid approach was demonstrated. This method used a period tracking method to predict a coarse wavelength shift, which was then fine-tuned to select the nearest peak in the interference spectrum to determine a much more precise change in the FPC length. Using a median filter to remove impulse noise, this hybrid method was demonstrated to produce consistent measurements with standard deviations on the order of nanometers, a hundred-fold lower than the precision obtained using similarly filtered period tracking methods. Furthermore, this precision is maintained while measuring cavity deflections >20 $\mu$m (>5% of the initial FPC length).
Funding
Office of Nuclear Energy.
Acknowledgments
This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). The authors would like to thank Dr. Brandon Wilson and Dr. Adrian Schrell for their valuable technical feedback regarding this work. This work was supported by the Versatile Test Reactor Program of the US Department of Energy, Office of Nuclear Energy.
Disclosures
DCS, AB, and CMP have submitted a patent application related to this work.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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