## Abstract

Recently it was shown that sinusoidal frequency scan optical frequency domain reflectometry (SFS-OFDR) can achieve remarkable performance in applications of distributed acoustic sensing (DAS). The main advantage of SFS-OFDR is the simplicity with which highly accurate sinusoidal frequency scans can be generated (in comparison with linear frequency scans). One drawback of SFS-OFDR has been the computationally intensive algorithm it required for processing of the measured backscatter data. The complexity of this algorithm was O(*N*^{2}) where* N* is the number of backscatter samples. In this work a fast processing algorithm for SFS-OFDR, with computational complexity O (*N *log *N*), is derived and its performance and limitations are studied in details. The new algorithm facilitated highly sensitive DAS operation over a sensing fiber of 64km, with 6.5m resolution and scan rate of 400Hz. The high sensitivity of the system was demonstrated in a field trial where it successfully detected human footsteps near the end of the fiber with excellent SNR.

© 2017 Optical Society of America

## 1. Introduction

Reflectometry in optical fibers refers to a host of methods which utilize backscattering in the fiber for distributed measurement of various physical parameters or signals [1]. Among these methods, those which enable distributed measurement of acoustically induced dynamic strain are of great interest in recent years [2]. This area of research is commonly called Distributed Acoustic Sensing (DAS). A DAS system is equivalent to an array of many microphones and is usually deployed over long distances (up to tens of kilometers). It can be deployed in various media: along pipes, along fences, underground, underwater or in buildings and structures. A DAS system typically comprises a sensing fiber connected to an interrogator. The interrogator transmits light into the fiber and detects the Rayleigh backscattered light. Interrogators can be categorized according to the domain in which the measured fiber profile is obtained. In Optical Time Domain Reflectometry [3] (OTDR) the interrogation waveform is a short pulse and the backscattered data is processed and presented in the time domain. In Optical Frequency Domain Reflectometry [4] (OFDR) the interrogation waveform is a long pulse or CW light whose instantaneous frequency varies linearly with time and the raw measurements are transformed to the frequency domain to yield the fiber's backscatter profile. Interrogation methods also defer in their detection principle. OTDRs utilize either direct detection [3] or coherent detection [5] while OFDR systems typically use coherent detection schemes [4,6]. Both approaches (including the one described in this paper) can detect multiple vibrations which occur simultaneously at different positions. The main advantage of OFDR over OTDR lies in the trade-off between sensitivity and spatial resolution: while in OTDR these two parameters are strongly coupled and improvement in one comes at the expense of the other, in OFDR it is possible to obtain both very high resolution and very high sensitivity [7–9]. The drawbacks of OFDR are mainly higher vulnerability to phase noise and the difficulty to produce highly linear frequency scans with high repetition rates [6,10]. Recently it was shown that the latter issue can be circumvented via Sinusoidal Frequency Scan Optical Frequency Domain Reflectometry [11] (SFS-OFDR). The excellent performance of SFS-OFDR stems mainly from the simplicity with which highly accurate sinusoidal frequency scans can be generated (in comparison with linear frequency scans). One drawback of SFS-OFDR has been the computationally intensive algorithm it required for processing of the measured backscatter data. The complexity of this algorithm was $\text{O}\left({N}^{2}\right)$where$N$ is the number of backscatter samples. This imposed a severe limitation on the performance of the method and prohibited its use in long-range real-time DAS applications. It also made comprehensive characterization of SFS-OFDR in simulations and experiments exceedingly time consuming and impractical. In previous work, a fast processing algorithm for SFS-OFDR [12], with computational complexity of $\text{O}\left(N\mathrm{log}N\right)$ was introduced. In this work, the underlying formalism of the new algorithm is derived and a thorough numerical study of the performance and limitations of SFS-OFDR is performed for the first time. Experimentally, the new algorithm was used to demonstrate DAS operation over a sensing fiber of 64km, with 6.5m resolution and at a scan rate of 400Hz. The high sensitivity of the system was confirmed in a field trial where it successfully detected human footsteps near the end of the 64km fiber with excellent SNR.

## 2. Theory

The underlying principles of SFS-OFDR were described in details in [11] but are briefly repeated here for completeness. In SFS-OFDR the instantaneous frequency of the laser is varied according to: ${f}_{inst}\left(t\right)={f}_{0}+0.5\Delta F\mathrm{sin}\left({\omega}_{r}t\right)$ where ${f}_{0}$ is the nominal laser frequency at $t=0$, ${\omega}_{r}=2\pi {f}_{r}$ is the modulation angular frequency and $\Delta F$ is peak-to-peak modulation amplitude. The laser output is split between a reference arm and a sensing arm, see Fig. 1. The backscattered light from the sensing arm is mixed with the reference and detected via a coherent I/Q detector. The complex response in a single scan can be expressed as [11]:

For example, the processing time in our lab PC (Intel Core i7-3770@3.40GHz with 16GB RAM) for raw data from a fiber simulation with sampling frequency of 200MHz, modulation frequency of 1KHz, and 15% processing window (i.e. 30000 samples) was 22msec for conventional OFDR, 500msec for Fast-SFS-OFDR and ~40sec for the old version SFS-OFDR.

A fast algorithm is especially desired in real time DAS applications. The new algorithm facilitated tests of SFS-OFDR which were previously impractical due to the excessive computational time they required. With the new algorithm the accuracy and performance of SFS-OFDR in various realistic scenarios was tested via computer simulations and lab and field experiments.

## 3. Implementation of SFS-OFDR

Our implementation of SFS-OFDR was described in details in [11]. It is briefly repeated here for completeness. It consists of an ultra-coherent 1550.12nm laser source (NKT Adjustik) whose instantaneous frequency is swept in a sinusoidal manner. The laser output is split between a reference arm and a sensing arm, see Fig. 1. The sensing arm directs the light into the sensing fiber using a circulator. An optical hybrid (by Kylia) is used to combine the backscattered light with the reference. Two balanced detectors with 400MHz bandwidth detect the optical hybrid's outputs. The data is sampled using an Agilent 9000A oscilloscope and processed with Matlab.

## 4. SFS-OFDR simulations

Similarly to conventional OFDR, the spatial resolution in SFS-OFDR is inversely proportional to $\Delta F$ [13]. It is therefore desired to capture and process at least a half of the scan period in order to exploit maximum range of frequency variations. It was shown, however, that the processing window cannot be broadened beyond a certain limit)that is bounded by 45%(due to the appearance of undesired artefacts in the recovered backscatter profile [11]. The artefacts are wide sidebands which accompany “legitimate” reflection peaks and may interfere and contaminate other parts of the backscatter profile. These artefacts are a result of the fact that the phase factors in Eq. (2) are not strictly orthogonal and their cross-correlation matrix contains elements with significant magnitude away from the main diagonal [11]. The need to avoid the sidebands means that for a given $\Delta F$ and ${f}_{r}$ the achievable spatial resolution and fiber length does not reach the fundamental limits $\Delta {z}_{\text{ref}}\equiv {v}_{g}/2\Delta F$ and ${L}_{\text{ref}}\equiv {v}_{g}/2{f}_{r}$ respectively. Roughly stated, the sidebands and the fiber profile start to interfere as the roundtrip delay or the processing window become comparable to quarter of the scan period. The effect of increasing the processing window is demonstrated in Fig. 2. It shows the results of a SFS-OFDR simulation of 64km fiber with different processing windows. The Rayleigh backscatter profile is seen between 0 and 64km. Beyond 64km there are only artefacts (sidebands). It can be seen that a proper choice of parameters is essential to prevent penetration of the sidebands to the backscatter profile.

The development of the fast SFS-OFDR algorithm enabled a comprehensive numerical study of the performance limits imposed by the presence of the sidebands. To implement the simulation the sensing fiber was represented as a concatenation of small backscattering sections of length $\delta z=1m$. The value of $\delta z$ was chosen to be smaller than the expected spatial resolution of the measurement. For each section a complex backscatter coefficient, ${R}_{k}$, was drawn from a normal distribution with zero mean and real and imaginary variances ${\sigma}_{r}^{2}={\sigma}_{i}^{2}={\sigma}^{2}=0.5\cdot {10}^{-4}$. The loss of the fiber was taken into consideration by multiplying each coefficient with its appropriate decay term: ${R}_{k}\mathrm{exp}\left(-2\alpha {z}_{k}\right)$ were $\alpha =0.046\text{\hspace{0.17em}}{\text{km}}^{-1}$. To enable estimation of the spatial resolution a 4% reflection peak was added at the fiber's end. The spatial resolution, $\Delta z$, was defined as the Full Width Half Maximum (FWHM) of the end peak. Figure 3(a) shows results of a numerical characterization of the spatial resolution as a function of the fiber length, $L$, for different scan rates, ${f}_{r}$. The width of the processing window was set to 15% of the scan period. The graphs show normalized spatial resolution namely, $\Delta z/\Delta {z}_{\text{theory}}=\Delta z/\left({v}_{g}/2\Delta {F}_{15\%}\right)$ where $\Delta {F}_{15\%}\equiv \Delta F\mathrm{sin}\left(0.15\pi \right)$ is the total frequency range in a 15% window centered at the origin. It can be seen that all graphs start at $\Delta z/\Delta {z}_{\text{theory}}\approx 2.2$ for short fibers. This is due to the use of a Blackman window as part of the application of the fast-SFS algorithm (see Appendix A) [11]. It can also be seen that the optimal resolution deteriorates as the fiber length and the scan rate increase. The source of this behavior is the gradual increase in the use of the nonlinear parts of the sinusoidal scan, which enter the processing window, as the fiber length and/or the scan rate increase. As a result the effective frequency range decreases and so does the spatial resolution. In addition, for each ${f}_{r}$, the fiber length, $L$, was increased to the maximum value possible, ${L}_{\mathrm{max}}$, before the sidebands started penetrating the fiber's backscatter profile. In quantitative terms, ${L}_{\mathrm{max}}$ is the length where the skirt of the undesired sidebands at the fiber end reached a threshold of 50dB below the Rayleigh level at this position. In Fig. 3(b) ${L}_{\mathrm{max}}$ as well as ${L}_{\text{ref}}$ are given for each scan frequency. Clearly, spatial resolution can be improved by using more than 15% of the scan period but this, as well, is limited by the penetration of the sidebands. Figure 4(a) shows the maximum processing window achievable for each scan frequency, ${f}_{r}$. Again, the graphs end when the 50dB threshold is reached. The optimal spatial resolutions, which can be achieved with the maximum processing windows, are listed for each scan frequency and fiber length in Fig. 4(b). The values in the table are normalized by $\Delta {z}_{\text{theory}}\equiv {v}_{g}/2\Delta {F}_{x}$ where $\Delta {F}_{x}$ is the total frequency range for a window of size $x$.

## 5. Experiments

The simulation enabled a judicious choice of parameters in the experimental study of the method. Two experiments were performed, one aimed to characterize the spatial resolution of SFS-OFDR as a function of length. The second was a field trial which tested the use of the method for sensing of dynamic strain at very long distances. As in the simulation, spatial resolution was characterized by placing a discrete reflector (an FC/PC connector) at the end of the sensing fiber and determining its FWHM in the recovered backscatter profile. The sensing fiber was made of a concatenation of spools of telecom type single mode fiber. Four sensing fibers of total lengths 1, 13, 39.2 and 64.6 km were characterized, seen in Fig. 6(a). The scan rate was set to 300Hz and a 15% processing window was used (i.e. 500μs). An example of the outcome of the Fast-SFS algorithm in the case of the 64.6km fiber is shown in Fig. 5. Figure 6(b) comprises plots of the experimentally obtained spatial resolutions (red stars) as well as the values obtained in a parameter-matched simulation (dashed gray). The discrepancy between the experimentally obtained spatial resolutions and the simulation results it attributed to the presence of high harmonics in the, ideally sinusoidal, instantaneous frequency of the laser [11]. It was found that by adding a second and third harmonics to the simulated instantaneous frequency, with amplitudes $-5\cdot {10}^{-5}$ and $1.1\cdot {10}^{-5}$ respectively (relative to the first harmonic), a good fit with the experimental results can be obtained (blue graph in Fig. 6(b)).

Next, the performance of SFS-OFDR in dynamic strain sensing was tested in a field experiment. A 63.97km sensing fiber was connected to the interrogator. The sensing fiber comprised ~63.80km of spools and patch-cords in the lab and in the corridors of the EE building, ~70m of cabled fiber buried in a 0.5m deep trench in the backyard and another 100m spool with a FC/PC connector at its end, see Fig. 7. The scan frequency was set to 400Hz, the sampling frequency was 500MHz and the processing window was 10% of the scan period, namely 250μs. The processing time of a single scan on our lab PC (Intel Core i7-3770@3.40GHz with 16GB RAM) was ~2 seconds, which is not yet real-time but far more efficient than the original implementation which was not practical to compute. It is expected that real-time operation will be enabled with more efficient use of computational resources and better choice of operation parameters. According to the simulation the theoretical spatial resolution with these parameters was 4.27m (in the absence of high harmonics in the instantaneous frequency). The measured spatial resolution was found to be slightly bigger than that as expected: 6.37m. The system was used to record a person (<60kg) walking in the vicinity of the buried fiber (horizontal distance ~1m). Fig. 8 shows a normalized waterfall of the fiber's profile after Fast-SFS processing. The dashed red box designates the response of the buried fiber. A zoom-in to this response is shown in Fig. 9(a). The recorded footsteps are enclosed by the black ellipse.

Detection of dynamical signals in the power backscatter profile is often challenging but it can be performed rather efficiently with appropriate signal processing [4]. Alternatively, using the differential phase between spatial samples provides better contrast for detection of dynamical signals [14,15]. It also provides quantitative and linear measurement of strain. Figure 9(b) shows the differential phase between spatial samples separated by 0.4m, filtered using a band-pass filter in the range 5 to 65Hz. The footsteps are evident with better contrast and the small range of most values in the resulting matrix (−0.03 to 0.03 radians) indicates negligible effect of phase noise despite the long distance. Figure 10 shows the phase difference signal at 63.85km spatial cell, which is in the buried section of the sensing fiber (in blue). For spatial steps of ~0.4m, $\lambda =1550\text{nm}$ and $n\approx 1.5$ the magnitude of the phase perturbations translates to ~100 nanostrains and the noise level to ~10 nanostrains [16]. Also plotted is an audio recording of the same event using a microphone held by the walking person (in red). The signals were synchronized manually according to the first step. The differences in the timing of the next steps can be explained by the different propagation media and different distances between the position of the excitation and the sensor location.

## 6. Discussion

Using sinusoidal frequency scan in OFDR is simple and accurate but it necessitates a special processing algorithm of the raw data in order to obtain high spatial resolution. Previous implementation of the processing algorithm was exceedingly time-consuming. The fast implementation of the SFS-OFDR algorithm presented in this paper resolves this issue and renders SFS-OFDR suitable for real time applications. Using simulations the new algorithm was shown to reproduce fiber profiles which were obtained by the original method with negligible error. Nevertheless, proper implementation of the fast SFS-OFDR involves some subtleties that should be mentioned. The first originates from Eq. (A3), which describes zero-padding of the measured data to the full size of the scan period. This operation is required for the subsequent step in the derivation of the main result but it increases the number of processed samples to $\tilde{N}={T}_{r}{f}_{sample}$, where ${f}_{sample}$ is the sampling frequency and ${T}_{r}=1/{f}_{r}$ is the scan period, rather than $N=T{f}_{sample}$ (the original number of samples).

The second issue is the truncation of the Jacobi-Anger expansion in Eq. (A6). Since Bessel functions of the first type decrease rapidly for orders larger than the argument, this leads to negligible errors provided that$A=\pi \Delta F/{\omega}_{r}<\tilde{N}/2$, which simplifies to${f}_{sample}>\Delta F$. Note that this requirement is nothing but the Nyquist criterion for proper sampling of$\mathrm{exp}\left\{-iA\mathrm{cos}\left[{\omega}_{r}\left(t-\tau \right)\right]\right\}$.

Subject to these issues, the ability to process the backscattered signal in longer windows is the method’s strength. It enables better spatial resolution and sensitivity over greater distances and with higher update rates. The limiting values of these four parameters are, of course, mutually dependent. Here SFS-OFDR achieved 6.5m spatial resolution over a distance of 64km at an update rate of 400Hz. In [11] it enabled 0.66m spatial resolution of a distance of ~1.5km at an update rate of 21kHz. In both cases the method detected CW dynamical strain or strain impulses of magnitude of tens of nanostrains or less, with high SNR, even at the end of the sensing fiber.

We note that the consequences of using such large processing windows, are (a) the need for sufficient computer memory and processing resources, and (b) addressing the accumulated acoustic noise in the sensing fiber [4]. However, these concerns are manageable as attested by the enhanced performance of the DAS system, using the fast SFS-OFDR method, described in this paper.

## 7. Summary

A new algorithm for processing raw data from SFS-OFDR measurements was presented. The algorithm is implemented with two consecutive FFT operations and is thus computationally efficient. The new algorithm is particularly useful in real time applications of SFS-OFDR. It enables processing of longer sampling windows with larger scan frequency ranges and leads to improved range, resolution and update rate. The new algorithm facilitated comprehensive computer simulations of SFS-OFDR and long-range field experiments. The high performance of the method was demonstrated by detecting human footsteps at the end of a 64km fiber with 6.5m resolution and 400Hz scan rate.

## Appendix A - mathematical derivation

The first step for obtaining the result in Eq. (3) is to use the Jacobi-Anger expansion:

## Funding

Ministry of Science, Technology & Space, Israel (3-11830).

## Acknowledgment

This research was supported in part by the Ministry of Science, Technology & Space, Israel. The authors wish to thank their partners, Meir Hahami and Yakov Botsev, from DSIT Solutions for fruitful discussions.

## References and links

**1. **X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel) **12**(7), 8601–8639 (2012). [CrossRef] [PubMed]

**2. **A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. **87**(1), 011501 (2016). [CrossRef] [PubMed]

**3. **J. C. Juarez, E. W. Maier, K. N. Choi, and H. F. Taylor, “Distributed Fiber-Optic Intrusion Sensor System,” J. Lightwave Technol. **23**(6), 2081–2087 (2005). [CrossRef]

**4. **D. Arbel and A. Eyal, “Dynamic optical frequency domain reflectometry,” Opt. Express **22**(8), 8823–8830 (2014). [CrossRef] [PubMed]

**5. **Y. Lu, T. Zhu, L. Chen, X. Bao, and S. Member, “Distributed Vibration Sensor Based on Coherent Detection of Phase-OTDR,” J. Lightwave Technol. **28**, 3243–3249 (2010).

**6. **X. Fan, Y. Koshikiya, and F. Ito, “Centimeter-level spatial resolution over 40 km realized by bandwidth-division phase-noise-compensated OFDR,” Opt. Express **19**(20), 19122–19128 (2011). [CrossRef] [PubMed]

**7. **H. Barfuss and E. Brinkmeyer, “Modified Optical Frequency Domain Reflectometry with High Spatial Resolution for Components of Integrated Optic Systems,” J. Lightwave Technol. **7**(1), 3–10 (1989). [CrossRef]

**8. **K. Tsuji, K. Shimizu, T. Horiguchi, and Y. Koyamada, “Coherent optical frequency domain reflectometry for a long single-mode optical fiber using a coherent lightwave source and an external phase modulator,” IEEE Photonics Technol. Lett. **7**(7), 804–806 (1995). [CrossRef]

**9. **R. Passy, N. Gisin, J. P. von der Weid, and H. H. Gilgen, “Experimental and Theoretical Investigations of Coherent OFDR with Semiconductor Laser Sources,” J. Lightwave Technol. **12**, 1622–1630 (1994).

**10. **X. Fan, Y. Koshikiya, and F. Ito, “Phase-noise-compensated optical frequency domain reflectometry with measurement range beyond laser coherence length realized using concatenative reference method,” Opt. Lett. **32**(22), 3227–3229 (2007). [CrossRef] [PubMed]

**11. **E. Leviatan and A. Eyal, “High resolution DAS via sinusoidal frequency scan OFDR (SFS-OFDR),” Opt. Express **23**(26), 33318–33334 (2015). [CrossRef] [PubMed]

**12. **L. Shiloh and A. Eyal, “Fast Sinusoidal Frequency Scan OFDR for Long Distance Distributed Acoustic Sensing,” Proc. SPIE **10323**, 25 (2017).

**13. **W. Eickhoff and R. Ulrich, “Optical frequency domain reflectometry in single-mode fiber,” Appl. Phys. Lett. **39**(9), 693–695 (1981). [CrossRef]

**14. **H. Gabai and A. Eyal, “On the sensitivity of distributed acoustic sensing,” Opt. Lett. **41**(24), 5648–5651 (2016). [CrossRef] [PubMed]

**15. **A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase-OTDR,” Meas. Sci. Technol. **24**(8), 85204 (2013). [CrossRef]

**16. **A. Masoudi and T. P. Newson, “High spatial resolution distributed optical fiber dynamic strain sensor with enhanced frequency and strain resolution,” Opt. Lett. **42**, 290–293 (2017).