Abstract

A method for phase sensitive quasi-distributed vibration and acoustical sensing is presented. The method is based on double optical frequency domain reflectometry interrogation of a sensing fiber with an array of discrete weak reflectors. Two replicas of the interrogation signal are launched into the sensing fiber. The time delay between the replicas is equal to the roundtrip time between two consecutive reflectors. Each peak in the spectrum of the returning signal is made from a coherent addition of the reflections of two consecutive reflectors. Its magnitude is highly sensitive to the optical phase in the fiber segment between the reflectors. The system was used to detect and locate the fall of a paperclip from height of 40cm onto a sandbox where a 15cm segment of the fiber was buried. In a different experiment the system successfully detected and located minute vibrations at 440Hz that were induced by touching the fiber with a tuning fork.

© 2014 Optical Society of America

1. Introduction

Fiber optical reflectometry techniques for distributed and quasi-distributed intrusion detection, acoustical sensing and vibration sensing are attracting a lot of attention in recent years [114]. Many distributed systems make use of Rayleigh backscattering for obtaining local acoustical information. Quasi-distributed reflectometry sensing systems comprise arrays of discrete reflectors which are positioned along the fiber [15, 16]. In general, in all reflectometry sensing systems it is desired to increase the back-reflected power from a given resolution cell as much as possible in order to improve the Signal to Noise Ratio (SNR) at the receiver and consequently improve the sensitivity, range, dynamic range and bandwidth. In quasi-distributed sensing the backscattered power is typically higher than in Rayleigh based distributed fiber sensors. Hence, it is expected that the performance of quasi-distributed sensing systems will benefit from enhanced SNR compared with distributed systems. In both types of systems highly sensitive detection of the targeted signal (vibrations or acoustical waves) can be implemented by measuring the phase of the backscattered light in the fiber. To achieve that the measurement setup is designed to transform the externally induced phase variations into position resolved information. In a typical system the roundtrip phase associated with a given position in the fiber may accumulate over very large distances and is thus a rapid noise-like process. The optical phase variations that occur in a short fiber segment, however, vary much slower with time and represent the local acoustical excitation of the fiber. In Rayleigh-based Phase-Sensitive Optical Time Domain Reflectometry (Φ-OTDR) the local phase variations are detected by using a laser source whose coherence length is longer than the spatial length of the interrogation pulse [1, 2]. To enhance the SNR some authors used coherent detection [4, 5]. Additional enhancement in SNR can be achieved by increasing the energy of the interrogation pulse. In OTDR, however, fiber nonlinearities and the trade-off between the pulse energy and the spatial resolution limits the maximum useable pulse energy [17]. In Optical Frequency Domain Reflectometry (OFDR) this limitation is lifted as the spatial resolution is determined solely by the scan range of the instantaneous frequency [9, 13, 18]. It is thus expected that a combination of phase-sensitive OFDR interrogation and a quasi-distributed sensing fiber with discrete reflectors will enjoy very high SNR compared with most of the competing approaches. Extracting of local phase information in such a system, however, remains a challenge. One manner to obtain local phase variations is to filter out the phase modulation that is experienced by each reflector and to numerically differentiate it with respect to position [11]. The drawback of this approach is that it requires digitizing and numerically processing signals with very fast and noise-like phase variations. This task is usually challenging, it necessitates sophisticated phase unwrapping algorithms and is difficult to implement in real time. In this paper we describe a new method for real-time dynamic detection of local phase variations in a quasi-distributed sensing system. The method is based on a dynamical OFDR system which enables real time, long range, highly sensitive acoustic sensing at high sampling rate. The system uses a fast scanning laser and coherent detection scheme. To obtain the phase variations at localized short segments of the sensing fiber we employ double interrogation. A differential delay line produces two replicas of the interrogation signal. The delay between the replicas is tuned to be equal to the roundtrip delay between the discrete reflectors. The two replicas propagate in the sensing fiber and their reflections are coherently detected by the system's receiver. As common in OFDR the output of the receiver is Fourier transformed and the magnitude is displayed. Each spectral component in the transform corresponds to a different position along the sensing fiber. As will be shown in the following section the double-interrogation produces in the displayed output a beating term which is highly sensitive to the optical phase in the corresponding segment in the sensing fiber. As acoustical perturbations disturb this segment the beating term fluctuates in correlation with the acoustical signal. The method was used to detect and locate the fall of a paperclip, from a height of 40cm, onto a sandbox were a short segment of the fiber (15cm) was buried at a depth of 10cm. In another experiment the system measured and located vibrations at 440Hz that were induced in the fiber by gently touching it with a vibrating tuning fork for a short period of time.

2. Theory

Consider a fiber-optic system as described in Fig. 1. Light from a laser source is launched into a measurement arm and the back-reflected beam is mixed with a reference and detected by a coherent I/Q receiver. In OFDR the frequency of the laser is varied at a nominally constant rate, γ, over a frequency range Δf. The laser output field can be expressed as:

E(t)=E0exp[j(ω0t+πγt2)]
where ω0 is the nominal radial frequency at t=0 . In the most general case the measurement fiber can be described as a distributed reflector with z-dependent reflectivity r(z). The laser output is split into two interrogation lightwaves which are launched into the sensing fiber with a differential delay of τd. Hence the fiber is interrogated twice with two differently delayed versions of the frequency scanned input. Consider a single reflection in the fiber at position z. The back-reflected field from this point can be expressed as:
ES(t)=r(z)[E(t2zv)+E(tτd2zv)]
The back-reflected field is detected by a coherent receiver whose reference is taken from the frequency scanned laser. After combining the quadrature components at the output of the coherent receiver, the system response to a spatial impulse is obtained:
V(t)=ar(z){exp{j[2πγ2zvt+ϕ1(z)]}+exp{j[2πγ(2zv+τd)t+ϕ2(z)]}}
where ϕ1(z) and ϕ2(z) are the phases acquired by the two interrogation waves in the trip to point z and back. To obtain the response of a distributed sensing fiber we integrate (3) over the entire fiber:
V(t)=a0Lr(z'){exp{j[2πγ2z'vt+ϕ1(z')]}+exp{j[2πγ(2z'v+τd)t+ϕ2(z')]}}dz'
Taking the Fourier transform of V(t) gives:
V˜(ω)=a[r˜1(ω)+r˜2(ω)]
where:
r˜1(ω)=r(z)exp[jϕ1(z)]|z=ωv/4πγr˜2(ω)=r(z)exp[jϕ2(z)]|z=ωv/4πγτdv/2
The magnitude of V˜(ω) is given by:
|V˜(ω)|2=a2{|r˜1(ω)|2+|r˜2(ω)|2+2|r˜1(ω)||r˜2(ω)|cos[ϕ1(ωv4πγ)ϕ2(ωv4πγvτd2)]}
In practice it is expected that |r˜1(ω)| and |r˜2(ω)| will remain constant from scan to scan. Hence the magnitude of V˜(ω) will fluctuate due to the variations of the phase difference in the cosine function in Eq. (7). Since both ϕ1(ωv/4πγ) and ϕ2(ωv/4πγvτd/2) are roundtrip phases their difference describes the phase in a localized fiber segment starting at z=ωv/4πγ and ending at z=ωv/4πγvτd/2. This signal in (7) is recorded for each position along the fiber and for each scan and constitutes the output of the sensor.

 

Fig. 1 the experimental setup.

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3. Experiment

The experimental setup is described in Fig. 1. Light from an ultra-coherent laser (Orbits Lightwave Ethernal) was split between a reference arm and a sensing arm. The laser instantaneous frequency was controlled by applying electric signal to internal piezoelectric actuators (PZT). To prevent undesired mechanical transients due to discontinuities in the driving signal the PZT was driven by a sinusoidal signal away from its resonant frequency. This resulted in a smooth variation of the lasers frequency. The period of the sinusoidal signal was around 185μs. Out of the total period a sub-segment of ~6μs, where the variation of the laser's frequency was close to linear, was recorded in each cycle. This corresponded to a frequency scan range of ~50MHz. The reference light beam was transmitted through a variable optical attenuator (VOA) followed by a polarizer into the reference port of a Dual Polarization 90° Optical Hybrid (90°OH). The light in the sensing arm was transmitted through a differential delay module and directed to the input port of a circulator (port 1). Thus, at port 2 of the circulator, where the sensing fiber was connected, there were two replicas of the interrogation signal with two different delays. The length difference in the differential delay module was chosen to be 20m which is equal to twice the spacing between consecutive reflectors. The returning light from the sensing fiber exited port 3 of the circulator and entered the signal port of the 90°OH. The 90°OH had four pairs of output fibers. Each pair of fibers was connected to a balanced optical receiver. These produced four electronic signals which represented the four degrees of freedom of the light returning from the sensing network. Hence, the described setup was capable of a complete measurement of the returning field (four degrees of freedom: two quadratures of two polarization components). In this work, however, we focused on the signals produced by one polarization component. The polarization dependent attributes of the methods are the topic of a further study.

The outputs of the balanced receivers were sampled, analyzed and displayed by a Digital Storage Oscilloscope (600MHZ, Agilent Infiniium MSO9064A).

A sensing fiber with an array of discrete reflectors was connected to the interrogation unit. The array comprised 10 weak Fiber Bragg Gratings (FBG) (custom made by Fibertronix), with equal spacing of 10m. All FBGs had the same reflectivity, 0.5% and center wavelength, 1550.12nm. The full width half max (FWHM) bandwidth of all the grating was ~3nm (~375GHz). Their typical temperature shift was 10pm/C° (1.25GHz/ C°). The FBGs were designed to give strong uniform reflection relative to the Rayleigh backscatter (>30dB) over a frequency range which was much broader than the scan range. Accordingly, the system tolerance to undesired wavelength shifts of the FBG’s was very high. The total length of the fiber (including two long patch-cords) was ~940m. The first FBG was located roughly 360m away from the interrogator. A 15 cm segment of the fiber was buried in a box with slightly wetted sand at depth of 10cm (Fig. 2).

 

Fig. 2 impulse response with a free fall of a paper clip.

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To characterize the impulse response of the system a paperclip was dropped onto the sand from a height of 40cm. The frequency scan rate was 5.41kHz leading to acoustical bandwidth of ~2.7kHz. In another characterization experiment the response of the system to vibrations was tested by touching a segment of the fiber gently, for a short time, with a vibrating tuning fork (440Hz). In another characterization experiment, as shown in Fig. 3, the response of the system to vibrations was tested by touching a segment of the fiber gently, for a short time, with a vibrating tuning fork (440Hz).

 

Fig. 3 Vibration excitation with a A440Hz tuning fork.

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4. Results

Examples of the system output as a function of time are shown in Fig. 4. The top plot was generated with one arm of the differential delay open (no double interrogation) and the bottom plot shows results of the same measurement in the case of double interrogation. In both cases the plots were generated by recording the relevant time-segments in the system output over a period of 3 seconds. Each (time) segment was multiplied by a Hanning window and was Fourier-transformed. During the measurements a paperclip was dropped over a short section of the second fiber segment (between the second and the third reflectors) as described in sec. 3. Figure 4 shows the square magnitudes of the Fourier transforms in dB scale (10log10|V˜(ω)|2) as a function of time. The plots shows the backscatter profile from the central part of the sensing fiber where the FBGs were located. The ten peaks corresponding to the array of FBGs are clearly seen in the top plot. In the bottom plot each peak is doubled due to the double interrogation. Since the double-interrogation differential-delay matches the spacing between the FBGs the bottom plot comprises 11 peaks. The central 9 peaks are each a result of a coherent addition of reflections from two consecutive FBGs.

 

Fig. 4 Backscatter response with single interrogation (top) and double interrogation (bottom).

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As described above the main effect of external excitation is to perturb the phase in the beating term in Eq. (7). The effect of the phase variations on the peaks amplitude is immediately visible on the oscilloscope screen when it is set to display the magnitude of the signal FFT. The peaks amplitude becomes instable and starts to fluctuate in time. Hence, any recorded response of the peaks intensities (such as the one in the bottom plot of Fig. 4) shows time variations as well as unequal means. This is apparent in Fig. 4 as the peaks height in the bottom plot varies with time while the peaks in the top plot maintain constant height. The impulse response due to the fall of the paperclip is visible in the third backscatter peak (that we term sensor 2) in the double interrogation results. A clearer demonstration of the response is shown in Fig. 6 below. The graphs in Fig. 5 present another manifestation of the phase sensitivity of the double interrogation method. To generate this plot the magnitudes of the backscattered peaks at their centers, as a function of time, were extracted from the data. The resulting time varying signals were spectrally analyzed and their Power Spectral Densities (PSD) were calculated. The plots are averages of all relevant responses where in the case of double interrogation the edge peaks as well as the signal of the (excited) third peak were excluded. It can be seen that the spectral response corresponding to double interrogation is much higher than the response of single interrogation especially in the frequency range 0<f<100Hz. The peaks in 50Hz and its harmonics are common to both methods. These artefacts are attributed to interfering signal from the power supply that contaminates the voltage delivered to the laser tuning PZT.

 

Fig. 5 PSDs of the sensors outputs with single interrogation (blue) and double interrogation (red).

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Figure 6 represents the responses of the sensor array as a multichannel seismograph. The graphs are magnitudes of backscatter peaks, in the centers of the peaks, separated vertically to allow their presentation in the same plot. The response to the drop of a paperclip over a short section of the second fiber segment is undetectable in the single interrogation method but is remarkably strong in the double interrogation results. Besides the strong response in the second fiber segment crosstalk can also be observed in the other segments. The detailed mechanism of the crosstalk is currently under study. Its origin is attributed to the following effect. When a given position in the fiber is excited by an abrupt external excitation the phase of the light that passes through this position at the time of the excitation is modulated. This phase-modulated light continues to propagate in the fiber and to interrogate the fiber segments which are further away in the fiber (namely, segments on the left hand spool in Fig. 2). Although the phase modulation is common to both interrogation signals and its effect on the peaks magnitudes is significantly suppressed by the coherent addition, it yields residual responses in the unexcited segments. These residual signals can be further suppressed by proper high-pass filtering as demonstrated in Fig. 7. The price of such signal processing, however, is suppression of low frequency content in the sensors outputs.

 

Fig. 6 The impulse response of the system with single interrogation (a) and double interrogation (b).

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Fig. 7 The impulse response of the system with double interrogation after HPF.

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The system response to vibration excitation was tested by touching the fiber gently with a 440Hz tuning fork. The fork was brought in contact with the fiber for a period of ~3sec right after it was made to vibrate by banging it on a solid object. The PSDs corresponding to the 9 sensing segments are plotted in Fig. 8. Once again it can be seen that the results are consistent with the theory in sec. 2. Spectral peaks at 440Hz and its harmonics can be observed in the second fiber segment only in the results corresponding to the double interrogation.

 

Fig. 8 The system response to vibration excitation with single interrogation (a) and with double interrogation (b).

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5. Discussion

The performance of a distributed or quasi-distributed sensing system depends on many factors such as the SNR at the receiver, the type of the receiver and its sensitivity, coupling of the sensing fiber to the environment, use of mechanical resonators for sensitivity enhancement, signal processing of the measured signals, use of averaging and more [7,16,19]. Hence, a comprehensive comparison between the various methods is very difficult. It is, however, possible to compare certain parameters between measurement approaches assuming all other parameters are optimized. One of the most important parameter is the optical energy which is reflected from a resolution cell in a single scan period. This parameter will have a very significant effect on the performance of the method particularly on its sensitivity, dynamic range, range and bandwidth. Figure 9 describes the number of backscattered photons at the receiver for different systems and interrogation methods.

 

Fig. 9 Number of backscattered photons from a 10m resolution cell.

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In all plots it was assumed that the peak power of the input pulse was 10mW. The blue line represents the mean number of photons backscattered from a fiber segment of 10m due to Rayleigh scattering. It was calculated assuming a typical ratio between the powers of the interrogating pulse and the initial Rayleigh backscattered power: 10log10(Pbs/Pin)60 (for 10m spatial resolution which corresponds to pulse duration of 100ns). The number of backscattered photons in the case of Rayleigh backscattering and OFDR interrogation is plotted in green. The increase in backscattered photons is due to the use of longer pulse duration. The pulse duration was chosen to be 6μs as in the experiment. The red and cyan lines show respectively reflections from 10m spacing arrays of 0.1% reflectors and 0.5% reflectors in the case of OFDR interrogation with pulse duration of 6μs. It can be seen that quasi-distributed OFDR systems can offer orders of magnitude increase in the backscattered signal. To obtain spatial information from the measured data different signal processing will be employed in the different interrogation methods. To evaluate the SNR it is required to propagate the noise through these signal processing procedures. Due to the incoherent nature of the noise it is expected to increase as the square root of the pulse duration while the signal increases linearly with it [17]. Accordingly, significant improvement in the SNR is expected. This comes with a price of increased cost of the sensing fiber and an increase in the total attenuation. The latter factor, however, can be partly mitigated by optimizing the reflection parameter of the reflectors. It can be seen that for reflection coefficient of 0.01% significant increase in the number of backscattered photons exists even for 100km fibers. Another benefit of reducing the reflection coefficient is that it suppresses multiple reflections.

The spatial resolution of the proposed method is determined by FBG’s spacing. The minimum distance between the FBG’s (maximum achievable resolution) is determined by the fundamental spatial resolution of an OFDR system. This fundamental resolution is determined by the frequency scan range, ΔF, according to: Δzv/(2ΔF) where v is the light velocity in the fiber [18]. The scanning range in the laser that was used is roughly proportional to the driving voltage. With maximum supplied voltage it enables a tenfold improvement in spatial resolution compared with the resolution that was demonstrated but this will require a tenfold increase in the voltage supplied to the laser tuning mechanism (namely V>100V). Supplying such high voltages at sufficiently high frequencies is challenging. Hence, for dynamic sensing in the KHz range, the resolution of the system is currently limited to few meters.

The sensitivity of the system can be defined as the minimum acoustical or vibrational signal that can be detected. To characterize this parameter it is required to isolate the system acoustically and to produce a controlled acoustical excitation to one of its sensing segments. It is expected that the minimum detectable signal in these circumstances will be limited by the inherent phase noise of the laser and the phase noise induced by the amplitude noise of the voltage source that drives the laser tuning circuit. Another critical factor that affects the sensitivity of the system is the coupling quality between external acoustical field and in-fiber optical phase variations.

6. Conclusions

A dynamical OFDR technique for acoustical and vibration sensing was proposed and experimentally tested. The method was based on double interrogation of the sensing fiber. Two versions of the linearly chirped interrogation waveform were launched into the fiber with a delay of ~100ns between them. The returning signals were detected by a coherent receiver, sampled and analyzed. The spectral components fluctuated due to acoustically-induced phase variations in the sensing fiber. The system was excited with impulses and externally induced vibrations and successfully sensed the excitation in the correct segment while the single interrogation method failed.

References

1. 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]  

2. J. C. Juarez and H. F. Taylor, “Field test of a distributed fiber-optic intrusion sensor system for long perimeters,” Appl. Opt. 46(11), 1968–1971 (2007). [CrossRef]   [PubMed]  

3. Z. Y. Zhang and X. Y. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of Polarization-OTDR system,” Opt. Express 16(14), 10240–10247 (2008). [CrossRef]   [PubMed]  

4. X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, and J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010). [CrossRef]  

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

6. Z. G. Qin, T. Zhu, L. Chen, and X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011). [CrossRef]  

7. X. Y. Bao and L. Chen, “Recent Progress in Distributed Fiber Optic Sensors,” Sensors (Basel) 12(7), 8601–8639 (2012). [CrossRef]   [PubMed]  

8. M. M. Chen, X. P. Zhang, F. Wang, and X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

9. Z. Ding, X. S. Yao, T. Liu, Y. Du, K. Liu, Q. Han, Z. Meng, and H. Chen, “Long-range vibration sensor based on correlation analysis of optical frequency-domain reflectometry signals,” Opt. Express 20(27), 28319–28329 (2012). [CrossRef]   [PubMed]  

10. Z. G. Qin, L. Chen, and X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012). [CrossRef]  

11. O. Y. Sagiv, D. Arbel, Y. Katz, Y. Grotas, and A. Eyal, “Dynamical strain sensing via discrete reflectors interrogated by optical frequency domain reflectometry,” Proc. SPIE 8421, OFS2012 22nd International Conference on Optical Fiber Sensors, 84218L (2012). [CrossRef]  

12. H. Wu, J. Wang, X. Wu, and Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012). [CrossRef]  

13. D. P. Zhou, Z. G. Qin, W. H. Li, L. Chen, and X. Y. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express 20(12), 13138–13145 (2012). [CrossRef]   [PubMed]  

14. T. Zhu, Q. He, X. H. Xiao, and X. Y. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013). [CrossRef]   [PubMed]  

15. N. Linze, P. Tihon, O. Verlinden, P. Mégret, and M. Wuilpart, “Development of a multi-point polarization-based vibration sensor,” Opt. Express 21(5), 5606–5624 (2013). [CrossRef]   [PubMed]  

16. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997). [CrossRef]  

17. M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna Jr, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989). [CrossRef]  

18. P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, and N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000). [CrossRef]  

19. B. Culshaw and A. Kersey, “Fiber-optic sensing: A historical perspective,” J. Lightwave Technol. 26(9), 1064–1078 (2008). [CrossRef]  

References

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  1. J. C. Juarez, E. W. Maier, K. N. Choi, H. F. Taylor, “Distributed fiber-optic intrusion sensor system,” J. Lightwave Technol. 23(6), 2081–2087 (2005).
    [CrossRef]
  2. J. C. Juarez, H. F. Taylor, “Field test of a distributed fiber-optic intrusion sensor system for long perimeters,” Appl. Opt. 46(11), 1968–1971 (2007).
    [CrossRef] [PubMed]
  3. Z. Y. Zhang, X. Y. Bao, “Distributed optical fiber vibration sensor based on spectrum analysis of Polarization-OTDR system,” Opt. Express 16(14), 10240–10247 (2008).
    [CrossRef] [PubMed]
  4. X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
    [CrossRef]
  5. Y. L. Lu, T. Zhu, L. A. Chen, X. Y. Bao, “Distributed Vibration Sensor Based on Coherent Detection of Phase-OTDR,” J. Lightwave Technol. 28, 3243–3249 (2010).
  6. Z. G. Qin, T. Zhu, L. Chen, X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
    [CrossRef]
  7. X. Y. Bao, L. Chen, “Recent Progress in Distributed Fiber Optic Sensors,” Sensors (Basel) 12(7), 8601–8639 (2012).
    [CrossRef] [PubMed]
  8. M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).
  9. Z. Ding, X. S. Yao, T. Liu, Y. Du, K. Liu, Q. Han, Z. Meng, H. Chen, “Long-range vibration sensor based on correlation analysis of optical frequency-domain reflectometry signals,” Opt. Express 20(27), 28319–28329 (2012).
    [CrossRef] [PubMed]
  10. Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
    [CrossRef]
  11. O. Y. Sagiv, D. Arbel, Y. Katz, Y. Grotas, and A. Eyal, “Dynamical strain sensing via discrete reflectors interrogated by optical frequency domain reflectometry,” Proc. SPIE 8421, OFS2012 22nd International Conference on Optical Fiber Sensors, 84218L (2012).
    [CrossRef]
  12. H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
    [CrossRef]
  13. D. P. Zhou, Z. G. Qin, W. H. Li, L. Chen, X. Y. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express 20(12), 13138–13145 (2012).
    [CrossRef] [PubMed]
  14. T. Zhu, Q. He, X. H. Xiao, X. Y. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013).
    [CrossRef] [PubMed]
  15. N. Linze, P. Tihon, O. Verlinden, P. Mégret, M. Wuilpart, “Development of a multi-point polarization-based vibration sensor,” Opt. Express 21(5), 5606–5624 (2013).
    [CrossRef] [PubMed]
  16. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
    [CrossRef]
  17. M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
    [CrossRef]
  18. P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
    [CrossRef]
  19. B. Culshaw, A. Kersey, “Fiber-optic sensing: A historical perspective,” J. Lightwave Technol. 26(9), 1064–1078 (2008).
    [CrossRef]

2013 (2)

2012 (6)

D. P. Zhou, Z. G. Qin, W. H. Li, L. Chen, X. Y. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express 20(12), 13138–13145 (2012).
[CrossRef] [PubMed]

Z. Ding, X. S. Yao, T. Liu, Y. Du, K. Liu, Q. Han, Z. Meng, H. Chen, “Long-range vibration sensor based on correlation analysis of optical frequency-domain reflectometry signals,” Opt. Express 20(27), 28319–28329 (2012).
[CrossRef] [PubMed]

X. Y. Bao, L. Chen, “Recent Progress in Distributed Fiber Optic Sensors,” Sensors (Basel) 12(7), 8601–8639 (2012).
[CrossRef] [PubMed]

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
[CrossRef]

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

2011 (1)

Z. G. Qin, T. Zhu, L. Chen, X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[CrossRef]

2010 (2)

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

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

2008 (2)

2007 (1)

2005 (1)

2000 (1)

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

1997 (1)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

1989 (1)

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Askins, C. G.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Bao, X. Y.

Chen, H.

Chen, L.

X. Y. Bao, L. Chen, “Recent Progress in Distributed Fiber Optic Sensors,” Sensors (Basel) 12(7), 8601–8639 (2012).
[CrossRef] [PubMed]

Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
[CrossRef]

D. P. Zhou, Z. G. Qin, W. H. Li, L. Chen, X. Y. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express 20(12), 13138–13145 (2012).
[CrossRef] [PubMed]

Z. G. Qin, T. Zhu, L. Chen, X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[CrossRef]

Chen, L. A.

Chen, M. M.

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Choi, K. N.

Culshaw, B.

Davis, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Ding, Z.

Du, Y.

Foster, S.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Friebele, E. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Giffard, R. P.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Gisin, N.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Guinnard, L.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Guinnard, O.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Guo, H. X.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Han, Q.

He, Q.

Hong, X. B.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Huttner, B.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Juarez, J. C.

Kersey, A.

Kersey, A. D.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Koo, K. P.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

LeBlanc, M.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Li, W. H.

Li, Y.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Lin, J.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Linze, N.

Liu, K.

Liu, T.

Lu, Y. L.

Maier, E. W.

Mégret, P.

Meng, Z.

Moberly, D. S.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Nazarathy, M.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Newton, S. A.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Oberson, P.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Patrick, H. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Putnam, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

Qin, Z. G.

D. P. Zhou, Z. G. Qin, W. H. Li, L. Chen, X. Y. Bao, “Distributed vibration sensing with time-resolved optical frequency-domain reflectometry,” Opt. Express 20(12), 13138–13145 (2012).
[CrossRef] [PubMed]

Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
[CrossRef]

Z. G. Qin, T. Zhu, L. Chen, X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[CrossRef]

Ribordy, G.

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Sischka, F.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Taylor, H. F.

Tihon, P.

Trutna, W. R.

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

Verlinden, O.

Wang, F.

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Wang, J.

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Wang, X. C.

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Wu, H.

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Wu, J. A.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Wu, X.

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Wu, Y.

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Wuilpart, M.

Xiao, X. H.

Xu, K.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Yao, X. S.

Zhang, X. P.

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

Zhang, Z. Y.

Zhou, D. P.

Zhu, T.

Zuo, Y.

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Appl. Opt. (1)

IEEE Photon. Technol. Lett. (3)

P. Oberson, B. Huttner, O. Guinnard, L. Guinnard, G. Ribordy, N. Gisin, “Optical frequency domain reflectometry with a narrow linewidth fiber laser,” IEEE Photon. Technol. Lett. 12(7), 867–869 (2000).
[CrossRef]

Z. G. Qin, T. Zhu, L. Chen, X. Y. Bao, “High Sensitivity Distributed Vibration Sensor Based on Polarization-Maintaining Configurations of Phase-OTDR,” IEEE Photon. Technol. Lett. 23(15), 1091–1093 (2011).
[CrossRef]

Z. G. Qin, L. Chen, X. Y. Bao, “Wavelet Denoising Method for Improving Detection Performance of Distributed Vibration Sensor,” IEEE Photon. Technol. Lett. 24(7), 542–544 (2012).
[CrossRef]

J. Lightwave Technol. (5)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15(8), 1442–1463 (1997).
[CrossRef]

M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. 7(1), 24–38 (1989).
[CrossRef]

J. C. Juarez, E. W. Maier, K. N. Choi, H. F. Taylor, “Distributed fiber-optic intrusion sensor system,” J. Lightwave Technol. 23(6), 2081–2087 (2005).
[CrossRef]

B. Culshaw, A. Kersey, “Fiber-optic sensing: A historical perspective,” J. Lightwave Technol. 26(9), 1064–1078 (2008).
[CrossRef]

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

Microw. Opt. Technol. Lett. (1)

X. B. Hong, H. X. Guo, J. A. Wu, K. Xu, Y. Zuo, Y. Li, J. Lin, “An Intrusion Detection Sensor Based on Coherent Optical Time Domain Reflector,” Microw. Opt. Technol. Lett. 52(12), 2746–2748 (2010).
[CrossRef]

Opt. Express (5)

Sens. Lett. (2)

M. M. Chen, X. P. Zhang, F. Wang, X. C. Wang, “Development of Distributed Vibration Sensing System Based on Polarization Optical Time Domain Reflectometry,” Sens. Lett. 10, 1536–1540 (2012).

H. Wu, J. Wang, X. Wu, Y. Wu, “Real Intrusion Detection for Distributed Fiber Fence in Practical Strong Fluctuated Noisy Backgrounds,” Sens. Lett. 10(7), 1557–1563 (2012).
[CrossRef]

Sensors (Basel) (1)

X. Y. Bao, L. Chen, “Recent Progress in Distributed Fiber Optic Sensors,” Sensors (Basel) 12(7), 8601–8639 (2012).
[CrossRef] [PubMed]

Other (1)

O. Y. Sagiv, D. Arbel, Y. Katz, Y. Grotas, and A. Eyal, “Dynamical strain sensing via discrete reflectors interrogated by optical frequency domain reflectometry,” Proc. SPIE 8421, OFS2012 22nd International Conference on Optical Fiber Sensors, 84218L (2012).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

the experimental setup.

Fig. 2
Fig. 2

impulse response with a free fall of a paper clip.

Fig. 3
Fig. 3

Vibration excitation with a A440Hz tuning fork.

Fig. 4
Fig. 4

Backscatter response with single interrogation (top) and double interrogation (bottom).

Fig. 5
Fig. 5

PSDs of the sensors outputs with single interrogation (blue) and double interrogation (red).

Fig. 6
Fig. 6

The impulse response of the system with single interrogation (a) and double interrogation (b).

Fig. 7
Fig. 7

The impulse response of the system with double interrogation after HPF.

Fig. 8
Fig. 8

The system response to vibration excitation with single interrogation (a) and with double interrogation (b).

Fig. 9
Fig. 9

Number of backscattered photons from a 10m resolution cell.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

E( t )= E 0 exp[ j( ω 0 t+πγ t 2 ) ]
E S ( t )=r( z )[ E( t 2z v )+E( t τ d 2z v ) ]
V( t )=ar( z ){ exp{ j[ 2πγ 2z v t+ ϕ 1 ( z ) ] }+exp{ j[ 2πγ( 2z v + τ d )t+ ϕ 2 ( z ) ] } }
V( t )=a 0 L r( z' ){ exp{ j[ 2πγ 2z' v t+ ϕ 1 ( z' ) ] }+exp{ j[ 2πγ( 2z' v + τ d )t+ ϕ 2 ( z' ) ] } }dz'
V ˜ ( ω )=a[ r ˜ 1 ( ω )+ r ˜ 2 ( ω ) ]
r ˜ 1 ( ω )= r( z )exp[ j ϕ 1 ( z ) ] | z= ωv / 4πγ r ˜ 2 ( ω )= r( z )exp[ j ϕ 2 ( z ) ] | z= ωv / 4πγ τ d v /2
| V ˜ ( ω ) | 2 = a 2 { | r ˜ 1 ( ω ) | 2 + | r ˜ 2 ( ω ) | 2 +2| r ˜ 1 ( ω ) || r ˜ 2 ( ω ) |cos[ ϕ 1 ( ωv 4πγ ) ϕ 2 ( ωv 4πγ v τ d 2 ) ] }

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