1. Introduction

Distributed Acoustic Sensing (DAS) has been attracting a considerable amount of attention in recent years for various applications including intrusion detection, railroad monitoring, pipeline protection, seismic profiling, oil and gas well monitoring and more [1–8]. All current implementations of DAS are based on fiber-optic reflectometry. Most DAS methods obtain the position information from the time delay of the backscattered signal. The spatial resolution in these methods is determined by the duration of the interrogating pulse: $\Delta z=v_g{T}_p/2$. Hence, as in Optical Time-Domain Reflectometry (OTDR) [9, 10], there is a fundamental tradeoff between the spatial resolution and the Signal to Noise Ratio ($SNR\propto P_{in}{T}_p$). It was recently demonstrated that DAS can also be implemented via Optical Frequency Domain Reflectometry (OFDR) [11–13]. OFDR is a well-established method for measuring the reflection profile of an optical fiber [14–18]. The method is based on transmitting into the fiber light whose instantaneous frequency varies linearly with time during at least some fraction of the interrogation period. The backscattered light from the fiber is then mixed with a reference and detected. The detector output (the beat signal) is Fourier-transformed to yield spatial information. In contrast with time domain reflectometry, OFDR enables excellent spatial resolution alongside with high SNR. This is possible since in OFDR the resolution is determined solely by the frequency sweep range, $\Delta F$, independent of the duration of the interrogation pulse: $\Delta z=v_g/\left(2\Delta F\right)$. Thus, using longer and more energetic pulses, it is possible to increase the SNR without sacrificing spatial resolution [16–18]. Despite its clear advantages, the use of OFDR for implementation of DAS was rarely studied. As described below, one reason for this is the technical challenges which arise when attempting to generate light with highly linear frequency sweep at high repetition rates.

There are two principal approaches for generating linearly frequency-modulated light for OFDR. One approach makes use of an external electro-optic modulator [19–22]. The second imparts frequency variations onto the transmitted light by direct modulation of one of the laser's parameters (e.g. temperature, the central wavelength of its reflectors, an intra-cavity filter etc.) [23–25]. There is a fundamental difference between the two aforementioned modulation methods. In the case of the externally-modulated OFDR, the modulator must be driven by a signal generator whose RF frequency can be swept over a frequency range of at least the desired $\Delta F$. On the other hand, in the case of the directly-modulated OFDR, the scanned frequency range depends on the modulation signal’s peak-to-peak amplitude rather than its bandwidth. Furthermore, an OFDR system has two other important design parameters: the scan repetition rate $f_r=1/T_r$, and the sweep time $T_s\le T_r$. In DAS related applications high repetition rates are necessary in order to handle wide acoustic bandwidths. Naively it would seem that for a bandwidth of $f_{\max }$ setting the repetition rate to $f_r=2f_{\max }$ should be sufficient. However, this requirement alone is not enough. Since OFDR assumes that the fiber is stationary within the sweep time, $T_s$, the sweep of the desired range, $\Delta F$, must be completed during a time much shorter than $T_r={\left(2f_{\max }\right)}^{-1}$. In such cases the necessary RF-generator specifications for implementing externally-modulated OFDR can be rather challenging. In contrast, as demonstrated below, a directly-modulated OFDR system can be implemented with a simple every day RF generator

These trivial requirements from the RF generator, however, facilitated by the direct-modulation method, are not without their price. First, the implementation of directly-modulated OFDR requires the use of a fast-tuning tunable laser with efficient conversion from driving signal to frequency. This is, however, not a severe hurdle as many such lasers are currently readily available. The harsher problem is that in direct-frequency-modulation a highly linear sweep is difficult to achieve, since the tuning mechanism is often intricate and non-trivially depends on the driving signal. In many lasers, for example, the tuning mechanism is based on a PZT. In these cases the driving signal must change smoothly in time and the use of a sawtooth driving signal is undesired due to the long transient oscillations that follow a jump in the signal. Another way to describe this problem is if we regard the tuning mechanism as a linear system whose frequency response is highly non-flat. In such a case attempting to drive it with a broadband signal (such as sawtooth) consisting of more than one frequency will result in a severely corrupted modulation that will be significantly different than the driving signal. To deal with this issue it is natural to work with a sinusoidal driving signal [12,13]. Although this function is not linear on the whole, using a short enough time window, it can be approximated as linear. Nevertheless, this inherent non-linearity introduces many problems. First of all, since the linear approximation must hold for the backscatter from the end of the fiber, there is an inherent tradeoff between the fiber length and the interrogation period. Consequently, the spatial resolution and acoustic bandwidth are hindered as the fiber is lengthened. Second, since the linear approximation cannot hold with the same precision for different reflectors with different delays, the spatial resolution is not constant with respect to position along the fiber. Furthermore, due to the same reason, the conversion between frequency and position may depend on the position along the fiber as well. These problems, all caused by the non-linearity of the frequency sweep, can be corrected using a variety of recently introduced methods [21,26]. Yet, these methods require auxiliary interferometers or inline reflectors and make no use of the prior knowledge regarding the non-linearity, caused by the driving signal.

In this paper a novel processing method for directly-modulated sinusoidal modulation based OFDR system is introduced. The Sinusoidal Frequency Scan OFDR (SFS-OFDR) accurately corrects for the driving-signal’s non-linearity without the need for auxiliary or inline interferometers. Consequently, by using longer interrogation time windows SFS-OFDR allows higher spatial resolution and acoustic bandwidth compared to regular Fourier-transform based OFDR. Moreover, it is experimentally demonstrated that SFS-OFDR is capable of achieving $\Delta z=0.66\text{m}$ static spatial resolution at a sweep repetition rate of $f_r=21\text{KHz}$ and interrogation time window of $T_s\approx 12\mu \text{s}$ along a $L\approx 1.48\text{km}$ fiber. Namely, it produces over $4.7\cdot {10}^7$ resolution cells per second. Furthermore, it is also experimentally demonstrated that SFS-OFDR is capable of detecting and resolving two $\sim 1\text{KHz}$ perturbations $1.74\text{m}$ apart, as well as locate the fall of a paperclip with $~0.6\text{m}$ resolution. The paper is organized as follows: In sec. 2 we briefly describe the theory behind SFS-OFDR; Sec. 3 describes the experimental apparatus; The main results are shown in sec. 4, demonstrating SFS-OFDR’s superiority over regular OFDR when direct modulation is used; Also in sec. 4 we examine the limits of SFS-OFDR and present dynamic results; Finally in sec. 5 we discus some of the implications and open issues of SFS-OFDR.

2. Theory

Consider a fiber-optic system as described in Fig. 1. Modulated light from the laser source is launched into a sensing arm, and the backscattered beam is mixed with a reference and detected by an I/Q optical receiver. The frequency of the laser source is varied in a sinusoidal manner over a frequency range $\Delta F$, such that the instantaneous frequency is given by:

 

Fig. 1 The experimental setup.

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An approximated expression for the reflection profile can be extracted from the detected signal $V\left(t\right)$ by projecting it onto z-dependent phase factors corresponding to the expected phase of a backscatter from position $z$:

3. Experiment

An experimental OFDR system with an array of discrete reflectors was constructed (Fig. 1). Light from an ultra-coherent tunable laser source (NKT Koheras AdjustiK Benchtop) with a central wavelength of $1550nm$was split using a $50/50$ PM splitter between a reference arm and a sensing arm. The reference arm was connected to the Local Oscillator (LO) port of a Dual Polarization $90°$ Optical Hybrid (Kylia), while the second output of the PM splitter was connected to the sensing arm using an optical circulator. Port 2 of the circulator was used to connect the sensing arm, and the output of the circulator (port 3) was fed into the Signal port of the optical hybrid, which had four pairs of output fibers. Each pair of fibers was connected to a balanced optical receiver, which produced four electronic signals representing the four degrees of freedom of the light returning from the sensing arm. Hence, the described setup was capable of a complete measurement of the returning field (four degrees of freedom: two quadratures of two polarization components). These were sampled and stored by a Digital Storage Oscilloscope (600MHZ, Agilent Infiniium MSO9064A). The laser's instantaneous frequency was controlled by applying electric signal to an internal piezoelectric actuator (PZT) with a conversion coefficient of: $K\left(21\text{KHz}\right)=18.7\text{MHz}/\text{V}$. The PZT was driven by a sinusoidal signal with an amplitude of $10\text{V}$ and a frequency of $21\text{KHz}$, for a total frequency range (peak to peak) of $\Delta F\approx 373.5\text{MHz}$.

First, the sensing arm was connected to an array of 10 FBGs at equal center wavelengths, and $10\text{m}$ spacing. The FBG fiber was terminated with an open PC connector and its total length was $L\approx 814\text{m}$. Data was collected from within a half modulation period time window, centered on the expected backscatter delay of the first FBG. The collected data from a $3.57\mu \text{s}$ time window ($7.5\%$ of the modulation period) was analyzed using SFS-OFDR (according to Eq. (6)), as well as using the regular OFDR (Fourier-transform), in order to compare the methods for an approximately linear frequency modulation. Furthermore, the collected data from a $10.71\mu \text{s}$ time window ($22.5\%$of the modulation period) was analyzed to demonstrate SFS-OFDR’s ability to cope with the non-linear modulation. Second, the sensing arm was elongated by a $665\text{m}$ fiber to produce a total length of $L\approx 1479\text{m}$. Data was collected from a $12\mu \text{s}$ time window ($25\%$of the modulation period), centered on the sine’s zero crossing, and analyzed using SFS-OFDR (according to Eq. (6)) to achieve $0.66\text{m}$ FWHM static spatial resolution. Centering the time window on the sine’s zero crossing is necessary in order to get the maximal scanned BW for the given time window. Then, to demonstrate the dynamical capability, the sensing arm was connected to a $L\approx 780\text{m}$ fiber with two $<0.5\text{m}$ segments, $1.74\text{m}$apart, wrapped around PZT actuators. The first PZT was driven by a $1070\text{Hz}$ sinusoidal signal, while the second was driven by a $1030\text{Hz}$ sinusoidal signal. Data was collected from a $9.5\mu \text{s}$ time window ($20\%$ of modulation time period, and about $0.95\%$ of perturbation time periods) over $5000$ modulation periods (namely, a total time of $~238\text{ms}$). This was done 50 times to yield statistical information. Lastly, a paperclip was dropped on the same $L\approx 780\text{m}$ fiber, where the segment containing the PZT actuator was replaced by segment buried in sand. Data was collected from a $12\mu \text{s}$ time window ($25\%$ of modulation time period) over $5000$ modulation periods (namely, a total time of $~238\text{ms}$).

4. Results

A Fourier-transform was applied to the collected data from the $3.57\mu \text{s}$, and $10.71\mu \text{s}$ time windows ($7.5\%$, and $22.5\%$ of the modulation period respectively). These time windows were chosen such that one is within the linear approximation and the other is not. The data sets were multiplied by a Blackman window prior to the application of the transform to suppress sidebands. The slope of the frequency modulation at the center of the time window was used to map the beat frequency axis into a distance axis. The resulting reflection profile magnitude is shown in Figs. 2(a)-2(d). The effects of the sinusoidal modulation can be clearly seen even within the small time window (Figs. 2(a) and 2(b)). First, the spatial resolution is degraded with increasing distance from the first FBG for which the time window is optimally linear. Second, the distance between the first and last FBG is incorrect, $85.05\text{m}$ instead of the expected $90\text{m}$. Also, the end of the fiber appears at $z=753.5\text{m}$ instead of the expected $L\approx 814\text{m}$. Furthermore, in the case of the longer time windows for which the linear approximation does not hold (Figs. 2(c) and 2(d)), it is clear that the non-linearity of the modulation not only engenders the aforementioned inaccuracies but also severely distorts the reflection profile.

 

Fig. 2 Reflection profile magnitude obtained via OFDR on the data collected from (a) the $3.57\mu s$ time window, and (c) the $10.71\mu s$ time window. A zoom-in to the 10 FBGs for (b) the $3.57\mu s$ time window, and (d) the $10.71\mu s$ time window.

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On the other hand, the same data was analyzed with SFS-OFDR (according to Eq. (6) in the same time windows, again with a Blackman window. The resulting reflection profile magnitudes are plotted in Figs. 3(a)-3(d) superimposed on the Fourier-transform based OFDR results. The 10 FBGs are clearly better resolved for both time windows. Moreover, their width is independent of their distance from the first FBG, the total distance between the first and tenth FBG is $89.7\text{m}$, and the end of the fiber appears at $z=813.8\text{m}$, all in much better accordance with the actual sensing fiber parameters, as well as within the resulting spatial resolution (as will be reported later). Furthermore, the reflection profile magnitude shows no signs of distortion for the larger time window, but rather a sharpened better resolved result.

 

Fig. 3 Reflection profile magnitude obtained via SFS-OFDR on the data collected from (a) the $3.57\mu s$ time window, and (c) the $10.71\mu s$ time window. A zoom-in to the 10 FBGs for (b) the $3.57\mu s$ time window, and (d) the $10.71\mu s$ time window.

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To better evaluate the spatial resolution offered by SFS-OFDR, the collected data from various time windows was processed, again using a Blackman window. The resulting reflection profile magnitudes are shown in Fig. 4. The time windows are specified in terms of percentages out of the modulation period. Notice that in addition to the contracting width there appears to be a slight left-shift of the peak, as well as gradually growing side bands visible after passing the $15\%$ time window. We attribute these effects to higher harmonics in the modulation process. These higher harmonics are further addressed in the discussion, and ways of coping with them are suggested. In practice these effects pose an effective limit on the width of the time window. Nevertheless, since the spatial resolution depends on the scanned bandwidth, most of the spatial resolution improvement is gained between the $5\%$ and $15\%$ time windows for which these effects are almost unobservable. It is noteworthy that in order for linear approximation to hold, and OFDR spectrum to be undistorted, a maximum time window of about $4\%$ should be applied. For instance, the time window used to produce the spectrum shown in Fig. 2(b) was $7.5\%$, and the effects of the non-linearity are clearly visible.

 

Fig. 4 (a) One of the FBGs’ peaks as obtained by various time windows specified as percentages out of the full modulation period. (b) Zoom in on the peaks up to the $-3dB$ width.

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Figure 5 shows the FWHM as a function of the scanned bandwidth (top axis) and time window (as percentage out of the modulation period, bottom axis). The measured data was fit to the expected theoretical dependence up to a multiplication constant corresponding to an effective widening: $\Delta z=A{v}_g/\left(\pi \Delta F\right)$. The fit yielded a value of $A=2.7$. For reference, the figure also shows simulated results for OFDR and SFS-OFDR, both using a Blackman window, where the constant $A$ was found to be: $A=2.45$ for OFDR, and $A=2.55$ for SFS-OFDR. Although the actual fit does not achieve the simulated value, it does not fall far behind. Moreover, the overall tendency is as expected from theory, and predicts a spatial resolution of $<0.5\text{m}$ for a half modulation period time window, i.e. a full BW time window. It should be noted that the theoretical value of $A$, without a Blackman window is $A=\sin {\text{c}}^{-1}\left(1/2\right)\approx 1.9$ for the case of OFDR, and $A=J_0^{-1}\left(1/2\right)\approx 1.5$ for the case of SFS-OFDR with a half modulation period time window (see Appendix for details). The theoretical spatial resolution stated above assumes negligible laser phase noise. In all our results, we have considered time windows of the order of $~10\mu \text{s}$, which is far shorter than the coherence time of our ultra-coherent source ($>1\text{ms}$). Clearly, for a source with a broader linewidth (shorter coherence time) phase noise effects cannot be generally neglected.

 

Fig. 5 FWHM of one of the FBGs as a function of the scanned BW and time window, as percentage from the modulation period, for the measured data and simulated results.

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Figure 6 below shows the results of applying SFS-OFDR to data collected within a $12\mu s$ time window ($25\%$ of the modulation period), corresponding to $\Delta F=0.264\text{GHz}$ from a $L\approx 1479\text{m}$ long fiber. This time window was chosen since it is the longest for which the unwanted effects presented in Fig. 4 are still negligible. The average measured FWHM of the 10 FBGs was $0.66\text{m}$, and an example can be seen in Fig. 6(c), alongside a simulated theoretical result for this scanned frequency range. The resulting FWHM is in uncanny agreement with the fit from Fig. 5 which predicts a FWHM of $0.67\text{m}$, and also not far behind the theoretical limit of SFS-OFDR predicts a FWHM of $0.37\text{m}$. It can clearly be seen that even though a Blackman window was used in the SFS-OFDR process, there are visible sidebands. As stated above, we attribute this to higher harmonics in the modulation process. Nevertheless, these effects are at least $10\text{dB}$ lower than the peak, and for sensing fibers without very high reflectors will be very low. Also visible is a considerable widening of the end of the fiber, contrary to the high reflector in the middle of the fiber. We attribute this to residual imbalance between the I/Q channels which affects the response in the presence of negative beat frequencies, although further research is needed to conclude this. It should be noted that the overall downslope of the reflection profile magnitude is due to the detector’s bandwidth, and not a side-effect of SFS-OFDR.

 

Fig. 6 (a) Reflection profile magnitude obtained via SFS-OFDR on data collected from quarter modulation period time window, (b) a zoom-in on the 10 FBGs, and (c) one of the FBGs’ reflective magnitude as measure, as predicted from theory and as predicted from theory when using Blackman window. The dashed line marks the $-3\text{dB}$ points.

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Finally, the dynamical sensing capabilities of SFS-OFDR were tested in two ways: 1. the ability to detect and resolve two closely spaced sinusoidal excitations at acoustical frequencies was tested; 2. the ability to detect and localize an impulse perturbation. For the first test, the excitations were applied to the sensing fiber via a pair of PZT cylinders. The lengths of the wrapped fiber segments was $<0.5\text{m}$ and the length of the fiber between them was $1.74\text{m}$. To demonstrate the spatial resolution of the method for dynamic measurements each of the PZT's was supplied with an AC voltage of a different frequency ($1070\text{Hz}$ and $1030\text{Hz}$) and the system output was recorded during 5000 consecutive scans (namely during a total time of $~238\text{ms}$). For each scan a window of $9.5\mu \text{s}$ ($20\%$ of the modulation period) was processed (according to Eq. (6)) yielding a time dependent reflection profile. This time window was chosen so that it would provide sufficient resolution to resolve the separate perturbations while not being adversely affected by the presence of higher harmonics (see Sec. 5 below). The dynamical information was obtained from the phase of the reflection profile [12,28]. First the phases of the 5000 consecutive profiles where referenced to the first one. Then, for each of the 5000 profiles, each resolution cell was referenced to the preceding resolution cell, effectively resulting in a differential-phase reflection profile. Next, the time-dependent differential-phase was extracted and Fourier-transformed to yield an acoustic spectrum for each resolution cell along the fiber. The dynamic spatial resolution can be estimated by plotting the magnitudes of these spectra, at the excitation frequencies ($1070\text{Hz}$ and $1030\text{Hz}$), as a function of position (Fig. 7). Since the fiber was exposed to ambient acoustic noise, in addition to the applied excitations, each recording yielded a slightly different graph. Hence, the graphs in Fig. 7 were generated from averages over 50 different measurements and from the corresponding standard deviations.

 

Fig. 7 (a) The magnitudes of the acoustical spectra at $1070\text{Hz}$ and $1030\text{Hz}$ as a function of position. Solid plots are averages over 50 measurements. Dashed plots are the averages plus and minus one standard deviation (b) Zoom-in on the location of the two perturbations.

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It is clearly seen in Fig. 7(a) that there are significant magnitudes at the excitation frequencies only at positions close to the perturbations. In Fig. 7(b) the two perturbations are clearly separated, even with respect to one standard deviation. The measured distance between the two peaks is $1.8\text{m}$ (with a $0.1\text{m}$ sample spacing), in good agreement with the actual spatial separation. At $20\%$ of the modulation period the expected static spatial resolution is $\Delta z\approx 0.81\text{m}$. It should be noted that the two peaks cross one another at $~-4\text{dB}$, suggesting they could indeed be even closer and still be separated, although not as close as $\Delta z\approx 0.81\text{m}$. This difference between the spatial resolution measured in the dynamical and static cases is partially explained by noting that the dynamical data was collected while the system operated with a $K\left(21\text{KHz}\right)=14.8\text{MHz}/\text{V}$ PZT conversion coefficient (rather than the $18.7\text{MHz}/\text{V}$ with which the static data was collected). Taking this into account we expect a static spatial resolution of $\Delta z\approx 1\text{m}$. Even so, it is clear that the dynamical spatial resolution does not achieve the static one, although it does not fall far behind. In fact, the dynamic spatial resolution is less than twice that of the static one, a major improvement on currently established methods [29]. For the second test, the fiber segment containing the PZT actuators was replaced by a $10\text{m}$segment from which $~5\text{cm}$were buried underneath $~1\text{cm}$of sand. A paperclip was dropped on the sand from a height of $~5\text{cm}$. SFS-OFDR (25% processing window) was used to obtain the time-dependent reflection profile of the sensing fiber (can be seen in Fig. 8(a)). For each position along the fiber the variance of the reflection profile was calculated within a moving time window of a $100$ modulation periods (a total time of $~4.76\text{ms}$). This variance as a function of position and time is shown in Fig. 8(b). The inset clearly shows increased variance due to the impact of the paperclip. The width of the detected peak implies $~0.6\text{m}$ dynamic spatial resolution in this case. Figure 8(c) shows the time dependence of the reflection profile at the position of the perturbation ($z\approx 671.5\text{m}$). Again the paperclip impact is evident.

 

Fig. 8 (a) Time-dependent reflection profile magnitude, (b) The variance of the reflection profile magnitude in a moving time window of 100 modulation periods with a zoom-in box on: $\text{z}=655-685\text{m}$; $\text{t}=0.12-0.14\text{s}$, and (c) Time-dependent reflection magnitude at the position of the perturbation.

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5. Discussion

In OFDR there are always nonlinearities entering the scanning mechanism, the most fundamental of which is the laser’s phase noise. When these nonlinearities are not accounted for they severely degrade the spatial resolution. Correctly accounting for these nonlinearities can be rather challenging, and requires auxiliary interferometers to accurately estimate them. In directly-modulated OFDR there is another, more dominant, cause of nonlinearities, which is the scanning mechanism itself. In SFS-OFDR the scan is intentionally performed in a sinusoidal manner. As long as the amplitude of the scan is small enough, the scan mechanism can be approximated as a linear time-invariant system. In contrast to attempting a sawtooth sweep, a sinusoidal profile contains a single frequency, making it the most natural choice for such a system. The method introduced in this paper models only the linear response of the laser, and achieves significant results, which were presented above. However, throughout the paper we have limited ourselves to analyzing data from within a quarter of the modulation period, even though the theoretical analysis predicts better results for larger time ranges. The reason for this seemingly unnecessary limitation is an observed degradation in the results, when using larger time windows. We believe these negative effects to be the consequence of higher harmonics in the scanning mechanism, which are not accounted for by the presented model. Figure 9(a) below shows the phase of a single reflector collected from a full modulation period time window. Notice the clear sinusoidal profile. Figure 9(b) shows the remaining phase after subtracting the first harmonic, i.e. the estimated phase according to the presented model. Indeed the presence of higher harmonics are clearly visible in the reflector’s remaining phase.

 

Fig. 9 (a) The phase of a single reflector, and (b) the remaining phase after subtracting the estimated phase according to SFS-OFDR.

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Figure 10 below shows the SFS-OFDR reflectivity-profile of this single reflector using data from half the modulation period superimposed on a simulated result of SFS-OFDR on a single reflector with second and third harmonics in its phase. The relative amplitude of these high harmonics in the simulation is $a_2=a_3=3\cdot {10}^{-3}$. Although such a relative amplitude might seem negligible, it is actually not, and the effects can clearly be seen in the figure. This is due to the fact that these high harmonics enter in the phase in Eq. (4).

 

Fig. 10 A comparison between the simulated effects of higher harmonics in the modulation process and the results obtained by SFS-OFDR on a half modulation period time window.

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We believe that it is possible to account for these high harmonics by generalizing the model, following the same steps as presented in the theory above.

On a different note, very little work has been done to quantify the bandwidth of OFDR based DAS, and also in this paper we have not addressed the issue in specifics. Nevertheless, as was already noted in the introduction, it is clear that in order to achieve a given bandwidth, using the Nyquist-imposed repetition rate is not enough. In this realization lies another advantage of SFS-OFDR. By using a sinusoidal internal modulation one fairly easily achieve very high repetition rates as well as short sweep times. It is also noteworthy that as far as SFS-OFDR is concerned working with very short sweep times need not impair the spatial resolution, since it is possible to keep the interrogation time constant and simply modulate with a higher frequency. For example, assuming that in order to have a bandwidth of $f_m=10\text{KHz}$ a sweep time of $T_s={10}^{-2}\cdot {\left(f_m\right)}^{-1}={10}^{-6}\text{s}$ is required, one can interrogate over $25\%$ of a $f_r=250\text{KHz}$ modulation period, or even $50\%$ of a $f_r=500\text{KHz}$ modulation period. These modulation frequencies are still fairly low and can easily be generated by everyday RF generators. It should be noted that the modulation frequency can reach values of $~250\text{KHz}$ with a $25\%$ time window still two orders of magnitude larger than the equivalent (spatial resolution wise) OTDR pulse. However, as can be seen in the Appendix, the weight function $g\left(z,z\text{'}\right)$ is periodic with a period of $v_g/\left(2f_r\right)$. This poses a limitation on the fiber length that becomes more severe as the repetition rate increases: $L<v_g/\left(2f_r\right)\approx {10}^8/f_r$. In practice, when attempting to reach this limit in the lab we have encountered a more severe limit which we believe is the result of the higher harmonics in the modulation process. Obviously the issue of bandwidth in OFDR and SFS-OFDR based DAS should be subject to further research.

6. Conclusion

In this work a novel method for implementing high spatial resolution DAS was studied. The method, Sinusoidal Frequency Scan OFDR (SFS-OFDR), has several significant advantages over competing methods for implementing DAS. First, as a variant of OFDR it offers superior SNR and spatial resolution in comparison with OTDR based methods. Second, in contrast with other OFDR systems, it can provide high spatial resolution alongside with high scan rates while maintaining trivial requirements from the needed RF generator and a simple design which does not require an auxiliary interferometer. The method experimentally achieved $0.66\text{m}$ static spatial resolution at a sweep repetition rate of $21\text{KHz}$ and interrogation time window of $~12\mu s$ along a fiber of $L\approx 1479\text{m}$, thus producing over $4.7\cdot {10}^7$ resolution cells per second. Furthermore, the method experimentally achieved a $-\text{}4dB$ separation between two $\sim 1\text{KHz}$ perturbations, $1.74\text{m}$ apart and located the fall of a paperclip with $\sim 0.6m$ resolution.

Appendix

By substituting Eq. (4) into Eq. (6) an integral form for $g\left(z,z\text{'}\right)$ is obtained:

(8)$$g\left(z,z\text{'}\right)=A{\displaystyle \underset{-\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\overset{\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\int }}\exp \left\{i\frac{\pi \Delta F}{{\omega }_r}\left[\cos \left({\omega }_r\left(t-\frac{2z\text{'}}{v_g}\right)\right)-\cos \left({\omega }_r\left(t-\frac{2z}{v_g}\right)\right)\right]\right\}dt}$$

Via a few algebraic manipulations it can be expressed as:

(9)$$g\left(z,z\text{'}\right)=A{\displaystyle \underset{-\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\overset{\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\int }}\exp \left\{-iH\left(z-z\text{'}\right)\sin \left[{\omega }_{r}t+F\left(z+z\text{'}\right)\right]\right\}dt}$$

where $H\left(z-z\text{'}\right)=\frac{2\pi \Delta F}{{\omega }_r}\sin \left[\frac{{\omega }_r}{v_g}\left(z-z\text{'}\right)\right]$, and $F\left(z+z\text{'}\right)=\frac{{\omega }_r}{v_g}\left(L-z-z\text{'}\right)$ are both independent of the integration variable.

First we may consider the special case where $T$ is chosen such that it is equal to the full modulation period. In this case the phase shift $F\left(z+z\text{'}\right)$ is irrelevant (the integrand is periodic over the integration range) and the integral is readily solved to give:

(10)$$g\left(z,z\text{'}\right)=\frac{A}{{\omega }_r}{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\exp \left[-iH\left(z-z\text{'}\right)\sin \left(x\right)\right]dx}=\frac{2\pi A}{{\omega }_r}{J}_0\left[H\left(z-z\text{'}\right)\right]$$

where $J_0\left[x\right]$ is zeroth order Bessel function of the first kind. Using the definition of two points being resolved if one's peak coincides with the other's first zero, as customary for OFDR, we obtain a theoretical spatial resolution of:

(11)$$\Delta z=\frac{v_g}{{\omega }_r}{\sin }^{-1}\left({\beta }_1\frac{f_r}{\Delta F}\right)\approx 0.77\frac{v_g}{2\Delta F}$$

where ${\beta }_1\approx 2.4048$ is the first zero of $J_0\left[x\right]$.

A more interesting case is when $T$ is chosen such that it is equal to the half the modulation period. It is a model case since it is the shortest time range to achieves the full offered $\Delta F$. In this case we consider the real and imaginary parts of $g\left(z,z\text{'}\right)$ separately. The real part of $g\left(z,z\text{'}\right)$ is actually periodic in half the modulation period. Thus, again the phase shift $F\left(z,z\text{'}\right)$ is irrelevant, and the integral is readily solved to give:

(12)$$\mathrm{Re}\left\{g\left(z,z\text{'}\right)\right\}=\frac{A}{{\omega }_r}{\displaystyle \underset{-\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\overset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\int }}\cos \left\{H\left(z-z\text{'}\right)\sin \left[x+F\left(z+z\text{'}\right)\right]\right\}dx}=\frac{\pi A}{{\omega }_r}{J}_0\left[H\left(z-z\text{'}\right)\right]$$

As for the imaginary part of $g\left(z,z\text{'}\right)$, via some algebraic manipulations it can be written as:

(13)$$\mathrm{Im}\left\{g\left(z,z\text{'}\right)\right\}=\frac{2A}{{\omega }_r}{\displaystyle \underset{0}{\overset{F\left(z+z\text{'}\right)}{\int }}\sin \left[H\left(z-z\text{'}\right)\sin \left(x\right)\right]dx}$$

This integral is not easily solved. Figure 11 below shows some simulations of $g\left(z,z\text{'}\right)$ for the case of $T=T_r/2$ as well as other cases. It is easy to see from the simulations that the imaginary part of $g\left(z,z\text{'}\right)$ is negligible in comparison to its real part. This can be understood by considering $F\left(z+z\text{'}\right)$ and $H\left(z-z\text{'}\right)$. First we note that typical repetition rates are ${\omega }_r-{10}^4\text{Hz}$, so ${\omega }_r/v_g\sim {10}^{-4}{\text{m}}^{-1}$, and $F\left(z+z\text{'}\right)\sim \left(L-z-z\text{'}\right)\cdot {10}^{-4}$. When $\left|F\left(z+z\text{'}\right)\right|\ll 1$, namely $\left(z+z\text{'}\right)\sim L$, we can expand the integrand to first order and find:

(14)$$\mathrm{Im}\left\{g\left(z,z\text{'}\right)\right\}\approx \frac{2A}{{\omega }_r}{\displaystyle \underset{0}{\overset{F\left(z,z\text{'}\right)}{\int }}\sin \left[H\left(z-z\text{'}\right)x\right]dx=}\frac{2A\left\{1-\cos \left[H\left(z-z\text{'}\right)F\left(z+z\text{'}\right)\right]\right\}}{H\left(z-z\text{'}\right){\omega }_r}$$

Since $H\left(z-z\text{'}\right)$ appears in the denominator, if it is very large the imaginary part is indeed negligible. If $H\left(z-z\text{'}\right)$ is very small then the numerator vanishes and again we find the imaginary part to be negligible:

(15)$$li{m}_{H\left(z-z\text{'}\right)\to 0}\frac{2\left\{1-\cos \left[H\left(z-z\text{'}\right)F\left(z+z\text{'}\right)\right]\right\}}{H\left(z-z\text{'}\right)}=0$$

When $\left|F\left(z+z\text{'}\right)\right|-1$, namely $\left(z+z\text{'}\right)\sim 0,2L$ and $L-{10}^4\text{m}$, then either both $z$ and $z\text{'}$ are close to zero or both are close to $L$. In any case $\left(z-z\text{'}\right)\ll {10}^4$ and we can expand $H\left(z-z\text{'}\right)$ to first order and find:

(16)$$H\left(z-z\text{'}\right)\approx \frac{2\pi \Delta F}{v_g}\left(z-z\text{'}\right)$$

Usually $\Delta F-{10}^9\text{Hz}$, so $\Delta F/v_g\sim 10{\text{m}}^{-1}$, and $H\left(z-z\text{'}\right)\sim 10\cdot \left(z-z\text{'}\right)$. In this case we find:

(17)$$\mathrm{Im}\left\{g\left(z,z\text{'}\right)\right\}\approx \frac{2A}{{\omega }_r}{\displaystyle \underset{0}{\overset{1}{\int }}\sin \left[10\cdot \left(z-z\text{'}\right)\sin \left(x\right)\right]dx}$$

For $\left(z-z\text{'}\right)\underset{˜}{>}1\text{m}$ the integrand oscillates rapidly, and thus the result is negligible. For $\left(z-z\text{'}\right)\underset{˜}{<}{10}^{-1}\text{m}$ the integral can be solved numerically to give $\mathrm{Im}\left\{g\left(z,z\text{'}\right)\right\}~2A/{\omega }_r$, which is not negligible in comparison to the real part. However, this is of no consequence, since for $\left(z-z\text{'}\right)\underset{˜}{<}{10}^{-1}$ we wish to obtain a large contribution to $g\left(z,z\text{'}\right)$.

 

Fig. 11 Simulated form of the weight function $g\left(z,z\text{'}\right)$ for (a) the case of $T=T_r/2$ and ${\omega }_r=2\pi \cdot 20\text{KHz}$, (b) the case of $T=T_r/5$ and ${\omega }_r=2\pi \cdot 20\text{KHz}$, and (c) the case of$T=T_r/5$ and ${\omega }_r=2\pi \cdot 50\text{KHz}$.

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Figure 11 depicts some simulations of $g\left(z,z\text{'}\right)$ with various parameters. For numerical reasons, in all the simulations we set $\Delta F=100\text{MHz}$ and $L=5km$. Figure 11(a) depicts the case of $T=T_r/2$ analyzed above and ${\omega }_r=2\pi \cdot 20\text{KHz}$. Figures 10(b) and 11(c) were calculated with $T=T_r/5$, as was used for the presented dynamical results. In Fig. 11(b) ${\omega }_r=2\pi \cdot 20\text{KHz}$, while in 10.d ${\omega }_r=2\pi \cdot 50\text{KHz}$. Notice that for $T=T_r/5$ (this is actually true for any $T<T_r/2$), $g\left(z,z\text{'}\right)$ rises to plateaus according to the value of $F\left(z+z\text{'}\right)$, and shows secondary maxima according to the value of $H\left(z-z\text{'}\right)$. However, these plateau and secondary maxima are still $20-30\text{dB}$ below the primary peak, and will not affect the Rayleigh reflection profile.

Acknowledgment

This research was supported in part by the Ministry of Science, Technology & Space, Israel.

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