Abstract

A real-time distributed optical fiber vibration sensing prototype based on the Sagnac interference in conjunction with the optical time domain reflectometry (OTDR) was developed. The sensing mechanism for single- and multi-points vibrations along the sensing fiber was analyzed theoretically and demonstrated experimentally. The experimental results show excellent agreement with the theoretical models. It is verified that single-point vibration induces a significantly abrupt and monotonous power change in the corresponding position of OTDR trace. As to multi-points vibrations, the detection of the following vibration is influenced by all previous ones. However, if the distance between the adjacent two vibrations is larger than half of the input optical pulse width, abrupt power changes induced by them are separate and still monotonous. A time-shifting differential module was developed and carried out to convert vibration-induced power changes to pulses. Consequently, vibrations can be located accurately by measuring peak or valley positions of the vibration-induced pulses. It is demonstrated that when the width and peak power of input optical pulse are set to 1 μs and 35 mW, respectively, the position error is less than ± 0.5 m in a sensing range of more than 16 km, with the spatial resolution of ~110 m.

© 2017 Optical Society of America

1. Introduction

Distributed optical fiber vibration sensor (DOFVS) has received much attention in recent years due to its advanced properties. It can detect and locate the external vibration continuously in a large range with high location accuracy [1], [2]. Hence, the sensor is suitable to be applied in leakage monitoring of oil and gas pipelines, health monitoring of civil and aircraft structures, perimeter security, and so on.

External vibration may induce the local changes of refractive index in the sensing fiber of a DOFVS, which could result in phase variation of propagating waves inside the fiber. Thus, interference or coherent detection techniques are always used in DOFVSs. Since the response feature of Sagnac interferometers to an external vibration is related to the vibration location, a number of reported optical structures and configurations are based on them, such as a single Sagnac interferometer [3], dual Saganc interferometers [4], [5], and combining the Sagnac interferometer with another type of interferometer [6], [7]. In addition, vibration location could also be obtained by calculating time interval between the vibration-induced phase modulations in two independent Mach-Zehnder or Michelson interferometers [8], [9]. In these DOFVSs, although there are many advantages such as high sensitivity, long spatial sensing range, low cost and so on, measurement of simultaneous multi-points vibrations is still quite a challenge and has not been well-solved.

Alternatively, the phase-sensitive optical time domain reflectometry (Φ-OTDR) provides a possible way to realize the location of simultaneous multiple-points vibrations [10–14]. In an Φ-OTDR-based DOFVS, highly coherent optical pulses from a laser with ultra-narrow line width are launched into a conventional single-mode fiber. Light reflected from different scattering centers interferes to produce the OTDR trace. A field test for intruders is demonstrated, achieving 19 km sensing length with 100 m spatial resolution [10], [11]. Later, for the detection of high vibration frequency in health monitoring of civil structure, the highest frequency response of 1 kHz in a sensing range of ~1 km with a spatial resolution of 5 m is achieved by using coherent detection [12]. After that, by using the modulated pulses, the maximum frequency response has been improved to ~3 MHz in 1064 m fiber [13]. The sensing distance of Φ-OTDR is also proved to be extended to >100 km, by adopting an optical amplifier [14]. However, performance of the DOFVS based on Φ-OTDR critically depends on the performance of the laser used in the sensor [10], [12]. Mirror instability of the laser such as frequency drift, polarization change, and spectrum variation could initiate strong Rayleigh scattering fluctuation, and deteriorate the system performance seriously. In addition, high cost of high-performance devices and complex structures used in the system also limits the practical engineering application of such DOFVS.

In this paper, a real-time distributed optical fiber vibration sensing prototype based on the Sagnac interference in conjunction with the optical time domain reflectometry (OTDR) [15], [16] was developed, in which, a wide-band optical source is used. Unlike the sensing mechanism of the proposed DOFVS based on Φ-OTDR, the coherent OTDR trace is produced by the Saganc interference of two light beams reflected by the same scattering center inside the sensing fiber. Hence, the sensor has a low requirement for the performance of light source. The sensing mechanism and performance of the DOFVS is studied thoroughly. Theoretical models for its response to single- and multi-points vibrations are established and analyzed. An abrupt and monotonous power change appears in the corresponding position of the OTDR trace when single-point vibration acts along the sensing fiber. As to the detection of multi-points vibrations, when the distance between the adjacent two vibration points is larger than half of the input optical pulse width, abrupt power changes on the trace induced by multi-points vibrations are separate and still monotonous. A time-shifting differential module is proposed, optimized and carried out to convert these monotonous changes to pulses for detection. An experimental setup is also built to verify the performance of the DOFVS. It is demonstrated that when width and peak power of the input optical pulse are taken as 1 μs and 35 mW, respectively, the sensing range is larger than 16 km and the spatial resolution is about 110 m. Better yet, the location error can be improved to less than ± 0.5 m, when the number of averaging times is increased to 32. Compared with the vibration sensing systems based on Φ-OTDR, our sensor has a simpler structure and lower cost. Thus, it has more potential for practical engineering applications.

2. Principles and sensing mechanisms

The scheme of the DOFVS based on the Sagnac interference in conjunction with the OTDR is shown in Fig. 1. A superluminescent diode (SLD) is adopted as the optical source to reduce the noise caused by the coherent interaction of Rayleigh backscattering signals from different scatterers in the fiber. Rayleigh backscattering signals generated in the optical circuit are captured by a balanced detector. The time shifting differential module is developed to measure the vibration position, which will be introduced later.

 

Fig. 1 Scheme of the DOFVS based on the Sagnac interference in conjunction with the OTDR; SLD: super luminescent diode; C1, C2: 2 × 2 directional couplers; LT: delay fiber.

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2.1 Optical interference mechanism

The optical pulse with the width, w, from the source is divided into two beams by the input coupler, C1. One is transmitted into the delay fiber, LT, and the other is sent to the following coupler, C2, directly. Then, the two pulse beams are coupled into the sensing fiber, LS, through the second coupler, C2. Each pulse beam is scattered back continuously in time domain when it is propagating in the sensing fiber. These backscattering waves are divided into two beams again by the second coupler, C2. Therefore, there are four routes for the optical waves transmitting from and back to the input coupler, C1: 1) Passing through LT and C2 to the sensing fiber, being scattered back inside the sensing fiber and returning through the same path; 2) Passing through LT and C2 to the sensing fiber, being scattered back but returning directly without passing through LT. 3) Passing though C2 to the sensing fiber directly, being scattered back inside the sensing fiber and returning through the same path; 4) Passing through C2 to the sensing fiber directly, being scattered back but returning through LT. Apparently, among them, only routes 2 and 4 have equal optical length. As the delay fiber is much longer than coherence length of the wide-band optical source, only the waves scattered by the same scattering center and propagating along routes 2 and 4 can interfere with each other in the input coupler, C1. Similar to Sagnac interferometer with the operation of continues wave [17], the waves propagating along route 2 can be defined as clockwise (CW) waves, while the waves propagating along route 4 can be called counterclockwise (CCW) waves.

Vibration acting at a certain position along the sensing fiber may change the refractive index of the fiber around that position, and this results in a phase shift, φ(t) to waves propagating through the position. It can be observed from the optical circuit that the waves backscattered behind the vibration position are modulated twice. When the waves in the form of a pulse propagate through the vibration position, their phases are modulated for the first time. After they are scattered back, they will be modulated for the second time when passing that position again. The time interval between the two phase modulations is determined by the distance between the scatterer and vibration positions:

ΔtV=2(lRlV)vg,
where lR is the scatterer position, lV is the vibration position, and vg is the velocity of optical light in the fiber core.

As to an OTDR system, the light power detected at a certain position is the power integration of the light backscattered from infinitesimal short intervals within the whole optical pulse [18]. When the width of the optical pulse is much smaller than the period of the vibration, vibration-induced phase changes of the light waves contributing to the power integration for a certain position, l, can be considered as the same. For instance, when width of the optical pulse and period of the vibration are 1μs and 1 ms, respectively, obviously the approximation can be made successfully. Thus, ΔtV for these waves can be approximated as

ΔtV(l)(llV)vg.
Similarly, vibration-induced phase changes of the light waves backscattered near the vibration position can also be treated by the same way.

The delay fiber, LT also provides a time interval between the phase modulations of CW and CCW waves, which can be given by

ΔtM=lTvg.

2.2 Single-point vibration sensing

When a single-point vibration loads at the position, lV, the vibration-induced phase difference between the CW and CCW waves backscattered at a certain position, l, can be described as

Δφ(l)={0l<lVφ(lVvg)+φ[lVvg+ΔtV(l)]φ(lVvg+ΔtM)φ[lVvg+ΔtM+ΔtV(l)]llV.
When l is relatively close to lV, Eq. (4) can be approximated as

Δφ(l)Δφ(lV)=2[φ(lVvg)φ(lVvg+ΔtM)].

The coherent OTDR trace produced by the interference of CW and CCW waves can be divided into three segments: 1) When fiber length, l, is less than lV, the received backscattering waves are not influenced by the vibration. 2) When fiber length, l, is bigger than lV but less than lV+w/2, parts of the received backscattering waves contributing to a power integration are phase-modulated; 3) When fiber length, l, is larger than lV+w/2, phase of all the received backscattering waves is perturbed by the vibration. Thus, power of the coherent OTDR signal detected at the output port A with fiber length, l, can be expressed as

PAC(l)={14ASP0[1exp(2αl)]exp(αlT)lw218ASP0[2exp(αw)2]exp[α(2l+lT)]w2<llv,18ASP0{2exp(αw){{1cos[Δφ(l)]}exp[2α(llV)]+{1+cos[Δφ(l)]}}}exp[α(2l+lT)]lv<llv+w218ASP0[exp(αw)1]{1+cos[Δφ(l)]}exp[α(2l+lT)]lv+w2<llS18ASP0{exp(aw)exp[2a(llS)]}{1+cos[Δφ(l)]}exp[α(2l+lT)]lS<llS+w2
AS=SαSαP0exp(αw)
where S is the capture coefficient of backscattering signal, αS is the scattering coefficient of the fiber, α is the attenuation coefficient of the fiber, P0 is the peak power of input optical pulse, w is the width of the optical pulse with the unit of length, lT is the length of the delay fiber, LT, and lS is the length of the sensing fiber, w/2<lV<lSw/2.

Correspondingly, power of the coherent OTDR signal detected at the output port B with fiber length, l, can be given by

PBC(l)={0llV18ASP0[exp(2αlV)exp(2αl)]{1cos[Δφ(l)]}exp(αlT)lV<llV+w2.18ASP0[exp(αw)1]{1cos[Δφ(l)]}exp[α(2l+lT)]lV+w2<llS18ASP0{exp(aw)exp[2a(llS)]}{1cos[Δφ(l)]}exp[α(2l+lT)]lS<llS+w2

It can be observed from Eqs. (6) and (8) that without any external vibration, power of the coherent OTDR signal detected at the output port A decreases exponentially with the increase in fiber length, as shown in Fig. 2(a) by solid blue line (PAC(l)), while power of the coherent OTDR signal detected at the output port B equals to zero due to the destructive interference, as shown in Fig. 2(a) by solid cyan line (PBC(l)). When the vibration is acting at the position, lV, with the increase of fiber length, l from lV to lV+w/2, light power detected at the output port A decreases more rapidly and an abrupt decline is formed on the trace, as shown in Fig. 2(a) by dashed red line (PAC'(l)), due to the increase of exp[2α(llV)] shown in the Eq. (6). On the other hand, when the fiber length, l is less than lV, phase of all the backscattering signals are not disturbed by the vibration, so power of the coherent OTDR signal detected at the output port B still equals to zero due to the destructive interference. With the increase of l from lV to lV+w/2, light power detected at the output port B rises significantly, as shown in Fig. 2(a) by dashed magenta line (PBC(l)), supported by the decrease of exp(2αl) in the Eq. (8). As to fiber length larger than lV+w/2, variations of the light power detected at the two output ports with fiber length are impacted by Δφ(t). When the fiber length, l, is relatively close to lV+w/2, Δφ(t) could be considered as a constant value, so light power detected at the two output ports decreases exponentially with the increase in fiber length, as shown in Fig. 2(a) by dashed red and magenta lines (PAC(l) and PBC(l)).

 

Fig. 2 (a) Coherent OTDR traces with and without vibration detected at the output ports A and B (Values of the parameters are from the book [18]). (b) Coherent OTDR traces with 2-points vibrations detected at the output port A.

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It can also be observed from the Eqs. (6) and (8), and Fig. 2(a) that width of vibration-induced abrupt and monotonous power change (decline or rise) on coherent OTDR traces equals to half of the input optical pulse length, w/2. Amplitude of these changes depends on width and peak power of the input optical pulse, and vibration-induced phase difference between the CW and CCW waves which is determined by vibration characteristics and the delay fiber length, as shown in Eq. (4).

Equations. (6) and (8) show the fact that all the light waves backscattered behind the vibration position are phase-modulated by the vibration. However, according to the OTDR theory, it is possible that multi-points vibrations could still be detected and located if their distance is long enough.

2.3 N-points vibrations sensing

Next, response of the optical circuit to N-points vibrations (N2) is studied. When the distance between any adjacent two vibrations is larger than w/2, based on the single-point vibration sensing model, power of the coherent backscattering waves at the output ports A and B with fiber length, l, can be expressed as

PAC(l)={14ASP0[1exp(2αl)]exp(αlT)lw218ASP0[exp(aw)1]{1+cos[(j=0iΔφj(l))]}exp[a(2l+lT)]w2<llV1(i=0)orlVi+w2<llVi+1(1i<N1)orl>lVN(i=N)18ASP0{{1+cos[j=0i1Δφj(l)]}exp(aw){{cos[j=0i1Δφj(l)]cos[j=0iΔφj(l)]}exp[2a(llVi)]+{1+cos[j=0iΔφj(l)]}}}exp[a(2l+lT)]lVi<llVi+w2(1iN)18ASP0{exp(aw)exp[2a(llS)]}{1+cos[(j=0iΔφj(l))]}exp[a(2l+lT)]lS<llS+w2
and
PBC(l)={18ASP0[exp(aw)1]{1cos[j=0iΔφj(l)}exp[a(2l+lT)]llV1(i=0)orlVi+w2<llVi+1(1i<N1)orl>lVN(i=N)18ASP0{{{1cos[j=0i1Δφj(l)]}exp(aw){1cos[j=0iΔφj(l)]}}exp(2al){cos[j=0iΔφj(l)]cos[j=0i1Δφj(l)]}exp(2alVi)}exp(alT)lVi<llVi+w2(1iN)18ASP0{exp(aw)exp[2a(llS)]}{1cos[j=0iΔφj(l)}exp[a(2l+lT)]lS<llS+w2,
respectively,where Δφ0(l)=0, which is used to simplify the equations, lVi is the position of the ith vibration, w/2<lVi<lV(i+1)<lSw/2, and Δφi(l) is the phase difference induced by the ith vibration.

Firstly, response of the sensor to 2-points vibrations of which the distance is small but larger than w/2 is investigated. When the vibrations change relatively slowly, for the fiber length, l, close to lV1, Δφ1(l) could be approximated as Δφ1(lV1), and for the fiber length, l, close to lV2, Δφ1(l) and Δφ2(l) could be replaced by Δφ1(lV2) and Δφ2(lV2), respectively. It can be observed from Eqs. (9) and (10) that abrupt power decline or rise can still be observed at the first vibration position on the coherent OTDR traces detected at the output port A or B. When fiber length is increased from lV1+w/2 to lV2, power of both the two OTDR signals decreases exponentially. The other abrupt and monotonous change induced by the second vibration appears at the position, lV2 on each OTDR trace. However, variation direction of the second abrupt change which is influenced by the first vibration could be positive or negative. For instance, if cos[Δφ1(lV2)]  is larger than cos[Δφ1(lV2)+Δφ2(lV2)] in Eq. (9), an abrupt decrease can be observed at the second vibration position on the coherent OTDR trace detected at the output port A, as shown in Fig. 2(b) by red dashed line (PAC'(l)). On the other hand, if cos[Δφ1(lV2)]  is smaller than cos[Δφ1(lV2)+Δφ2(lV2)], the second vibration induces an abrupt increase, as shown in Fig. 2(b) by blue dashed line (PAC"(l)). The coherent OTDR trace without vibration is also shown here by black solid line (PAC(l)) as a reference. It is clear that when the distance between the two vibration points is shorter than w/2, changes induced by these two vibrations will overlap. Therefore, spatial resolution of the optical circuit is w/2, the same as other typical OTDR systems [18]. However, it should be noted that the vibration position error relies on the location accuracy of vibration-induced abrupt and monotonous power changes on OTDR traces. Thus, position error could potentially be improved without reducing width of the input optical pulse.

Although the detection of any following vibration is influenced by all the previous ones, as to relatively low-frequency vibrations, Δφi(l) can be considered as a constant value over a certain range near each vibration position, so power changes induced by multi-points vibrations on coherent OTDR traces are still abrupt and monotonous. Hence, multi-points vibrations can be located by searching corresponding abrupt and monotonous changes on backscattering traces induced by them, when the distance between any adjacent two of them are longer than w/2. However, for the restoration of a specific vibration signal, influence of the previous vibrations should be considered.

3. Signal processing methods

3.1 Balanced detection

It can be observed from Eqs. (6) and (8), coherent OTDR signals detected at the output ports A and B are out of phase, so a balanced detector can be adopted to remove the common-mode background signal and noise, as shown in Fig. 1. The output signal of the balanced detector, V(l) can be described as

V(l)P(l)=kAPA-C(l)kBPB-C(l),
where P(l) is defined as the composite OTDR trace, kA and kB are gains for the OTDR signals detected at the output ports A and B, respectively.

3.2 Time-shifting differential module

In comparison with vibration-induced abrupt power changes on OTDR traces, the rest parts vary much more slowly with fiber length, as shown in Fig. 2(a) and 2(b). A time-shifting differential module shown in Fig. 1 is designed to convert the vibration-induced abrupt changes to pulses for easy detection and location. Output signal of the balanced detector is divided into two beams, of which one is delayed in the time shifting component and then sent to a subtractor, while the other is transmitted to the subtractor directly. The difference between the two beams, DV(l) is calculated by the subtractor:

DV(l)=V(l+TSvg2)V(l),
where TS is the delay time produced by the time shifting component.

Figure 3(b) shows the differential results of a composite OTDR trace with 2-points vibrations by green, blue and red lines when TSvg/2=50 m,  100 m and 200 m, respectively. Here, width of the input optical pulse, w, is set to 200m. The composite OTDR trace with 2-points vibrations is also shown in Fig. 3(a). It is true that vibration-induced abrupt power changes are converted to pulses by employing the differential module. Thus, vibrations along the sensing fiber can be located extremely easily by measuring peak or valley position of the pulses induced by them.

 

Fig. 3 (a) Composite OTDR trace with 2-points vibrations. (b) Differential results of composite OTDR trace for different delay times.

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It can also be observed from Fig. 3(b) that with the increase of delay time, the absolute amplitude of vibration-induced pulses increases first, when the delay time, TS increases to w/vg (TSvg/2=w/2), the absolute amplitude reaches its maximum value, and after that, it remains unchanged. When the differential module is used, spatial resolution of the sensor deteriorates and can be described as

RS=TSvg+w2=vg2(TS+Tw),
where Tw is the width of input optical pulse in time domain.

When the delay time, TS is less than w/vg, longer delay time indicates better resistance to the practical system noise, but poorer spatial resolution. Thus, a balanced value for TS need to be obtained experimentally.

4. Experimental setup

An experimental setup of the DOFVS based on the Sagnac interferometer in conjunction with the OTDR is established to verify its performance, as shown in Fig. 4. The center wavelength and band width of the SLD are 1550 nm and ~40 nm, respectively. The two couplers and delay fiber which are also very sensitive to vibration are kept in a vibration-isolated container. OTDR traces are sampled by a FBGA-based data acquisition (DAQ) card. Sample rate of the DAQ card is up to 100 MSa/s, so one sample value corresponds to 1 m spatial distance. The time-shifting differential module is implemented by hardware programming on the DAQ card. Two cylindrical piezo-electric transducers (PZTs) with 1 m fiber wounded are taken as vibration sources. The vibration intensity and frequency of PZTs are controlled by a function generator, and their positions can be adjusted by changing lengths of the sensing fiber coils used in the system.

 

Fig. 4 Experimental setup for the DOFVS based on the Sagnac interferometer in conjunction with the OTDR; LS1: the first sensing fiber coil; LS2: the second sensing fiber coil; LS3: the third sensing fiber coil, PZT: piezo-electric transducer.

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5. Results and discussions

Firstly, response of the DOFVS to a single-point vibration is tested. The first PZT is placed at ~6150 m and is driven by an electronic sine wave with the frequency of 1 kHz and the amplitude of 10 V (peak-to-peak value, VPP), while the other one is kept unworking. Here, peak power and width of the optical pulse are set to 35 mW and 1 μs (~200m), respectively. OTDR traces perturbed by vibration detected at the output ports A and B are shown in Fig. 5(a) and 5(b) by red lines, respectively. The output signal of the balanced detector (the composite OTDR trace) with vibration acting is also shown in Fig. 5(c) by red line. OTDR traces without vibration acting are also recorded and shown in these figures by blue lines as references. All the traces shown in Fig. 5(a), 5(b) and 5(c) are acquired directly without any averaging procedure. When the vibration loads, an abrupt increase and decrease appear at ~6150 m on the OTDR traces detected at the output ports A and B, respectively, and the power at the same position on the composite OTDR trace goes down rapidly, which are also supported by the theoretical model, as shown in Fig. 2(a). Width of these changes is measured to be 100 m, which equals half of the optical pulse length, w/2. It can also be observed from Fig. 5(b), without vibration, the OTDR signal detected at the output port B is no longer zero but decreases with fiber length, due to the contribution of incoherent backscattering waves. On the other hand, Fig. 5(a), 5(b) and 5(c) also show that by adjusting the ratio of gains for the two channels in the balanced detector, without vibration, exponentially-decreasing OTDR trace is compensated to be a horizontal line. Because of this, the spatial sensing range is verified to be extended to from 10 km [15] to > 16 km, later on.

 

Fig. 5 OTDR traces with and without vibration acting: (a) the OTDR traces detected at the output port A; (b) the OTDR traces detected at the output port B; (c) the output signals of the balanced detector (the composite OTDR traces).

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Next, the delay time, TS, of the time shifting component is optimized experimentally. Here, only the first PZT works. The composite OTDR traces without and with vibration acting, which are used as the input signals for the time-shifting differential module, are shown as an inset in Fig. 6. Time-shifting differential results of the composite OTDR traces with vibration acting for different delay times, Ts, of 0.5 μs, 1 μs and 2 μs are shown in Fig. 6 by blue, red and green lines, respectively. It can be observed that the vibration-induced abrupt decrease on the composite OTDR trace is converted to a negative pulse successfully.

 

Fig. 6 Time-shifting differential results of the composite OTDR traces when vibration acting for different delay times, TS, of 0.5 μs, 1 μs, and 2 μs.

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Variations of amplitude of the vibration-induced pulses with delay time, TS, for different input optical pulse widths, Tw, of 1 μs and 2 μs are shown in Fig. 7(a) by solid blue and red lines, respectively. With the increase of delay time, TS, amplitude of the pulses for both cases increases rapidly at first and then reaches a constant value when the delay time is larger than the optical pulse width, which is also supported by the results shown in Fig. 3 and 6. Compared with absolute amplitude, the signal to noise ratio (SNR) is a more reasonable parameter to be considered during the optimization. Here, the SNR of the sensor is defined as a voltage ratio of the maximum amplitude of vibration-induced pulses and the background noise level. Variations of the SNR with delay time, TS, for Tw = 1 μs and Tw = 2 μs, are shown in Fig. 7(a) by dashed red and blue lines, respectively. It can be observed that with the increase of delay time, TS, the SNR for both cases increases rapidly first, and then fluctuates slightly around 5.5 when TS is larger than 0.1 μs. Variations of width of vibration-induced pulses with delay time, TS, for different input optical pulse widths, Tw, of 1 μs and 2 μs are shown in Fig. 7(b) by red and blue lines, respectively. It can be observed that for both cases, with the increase of delay time, width of the vibration-induced pulses increases linearly, which is also supported by Eq. (17). Overall, with the decrease of delay time, TS, spatial resolution improves continually, but the SNR deteriorates rapidly when the delay time is too small. In our experimental setup, the optimal value of delay time is set to 0.1 μs. Thus, SNR and spatial resolution of the sensor are ~5.4 and ~110 m, respectively. Figure 8(a) and 8(b) give the vibration signals by superposing 400 adjacent composite OTDR traces and their time-shifting differential results, respectively.

 

Fig. 7 (a) Variations of amplitude of the vibration-induced pulse amplitude and the SNR with delay time, TS, for different optical input pulse widths of 1 μs and 2 μs. (b) Variations of the vibration-induced pulse width with delay time, TS, for different optical input pulse widths of 1 μs and 2 μs.

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Fig. 8 Detection of the vibration at ~6150 m: (a) 400 adjacent composite OTDR traces; (b) the time-shifting differential results of 400 adjacent composite OTDR traces (TS = 0.1 μs).

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The sensing system can be understood as a sampling process of the vibration with the sampling rate of repetition frequency of the input optical pulse. Figure 9(a) shows the time response at the vibration position of the senor to the vibration produced by the PZT driven by the sine wave with frequency of 500 Hz. It can be observed that frequency of the output signal of the sensor is twice as high as that of the vibration signal, because phase operating-points of the optical interferometer are 0 and π for the outputs A and B, respectively, similar to Sagnac interferometer [17]. The frequencies of the output signal at the vibration position when the sensing fiber is disturbed by the vibrations with frequencies of 500 Hz, 750 Hz and 1 kHz are 992 Hz, 1488 Hz, and 1984 Hz, respectively, as shown in Fig. 9(b). Thus, the frequency of the vibration can be detected through dividing the frequency of the response signal at the vibration position by 2.

 

Fig. 9 (a) Time response of the proposed DOFVS to vibration produced by the PZT driven by sine wave. (b) Frequency response of the sensor to vibration produced by the PZT driven by sine waves with different frequencies.

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After that, response of the sensor to multi-points vibrations is also tested. Both of the PZTs are connected to the function generator and driven by the electronic sine wave with the frequency of 1 KHz and the amplitude of 10 V. Here, the delay time of the time shifting component and the input optical pulse width are taken as 0.1 μs and 1 μs, respectively. Thus, distance between the two PZTs is set to 110 m. It is demonstrated that the composite OTDR signals decrease abruptly at the first vibration position, but they may decrease or increase abruptly at the second vibration position, as shown in Fig. 10(a). The composite OTDR trace with an abrupt decrease at the second vibration position is shown by red solid line (OTDR Trace 1), while the one with an abrupt increase at that position is shown by blue solid line (OTDR Trace 2). Their differential results are also shown in Fig. 10(a) by red and blue dashed lines, respectively. Figure 10(b) gives the vibration signals by superposing differential results of 400 adjacent composite OTDR traces for this case. It can be observed that by using the time-shifting differential module, abrupt changes induced by multi-points vibrations are converted to negative or positive pulses, and these changes and pulses are separate. Therefore, multi-points vibrations along the sensing fiber can be located by measuring the peak or valley positions of negative or positive pulses induced by them.

 

Fig. 10 Detection of 2-points vibrations whose distance is 110 m: (a) composite OTDR traces with the abrupt increase and decrease at the second vibration position and their differential results; (b) differential results of 400 adjacent composite OTDR traces.

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Although the spatial sensing resolution is limited by the optical input pulse width and the delay time of time shifting component, the sensing position error which are mainly caused by the system noise, could be improved by increasing the averaging times. Variations of the detected vibration position with observation time for different averaging times of 8, 16 and 32 are shown in Fig. 11 by green, blue and red lines, respectively. It can be observed that it is true that position error decreases when the number of averaging times increases, and when the number of averaging times is taken as 32, the position error is less than ± 0.5 m.

 

Fig. 11 Variations of the detected vibration position with the observation time for different averaging times of 8, 16 and 32.

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Next, the distance between the two PZTs is set to ~16 km to test the potential maximum sensing range of the sensor. Vibration signals by superposing differential results of 400 adjacent composite OTDR traces with the two vibrations are shown in Fig. 12. Although the maximum amplitude of vibration pulses at ~16430 m is much smaller than that of vibration pulses at ~509 m, the SNR around 16 km is still good enough. Spatial sensing range of the sensor is more than 16 km, and thus it is suitable to be applied in perimeter security which has a strong requirement for long spatial range.

 

Fig. 12 Detection of 2-points vibrations with the distance of ~16 km.

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6. Conclusions

The DOFVS based on the Sagnac interference in conjunction with the OTDR is analyzed theoretically and demonstrated experimentally in this paper. In the sensor, single-point vibration along the sensing fiber induces an abrupt decrease at the corresponding position on the detected composite OTDR trace. As to multi-points vibrations, although power change induced by following vibration is impacted by all the previous ones, they can still be located separately when the distance between any adjacent two is larger than half of the optical input pulse width. Time-shifting differential module is developed to convert vibration-induced abrupt changes to pulses for simplified positioning. Consequently, vibrations can be located accurately and easily by measuring peak or valley positions of the pulses induced by them.

A real-time experimental prototype for the sensor is built to verify the sensing mechanism and performance of the sensor. The experimental results are consistent with theoretical analysis. Pulse width and peak power of the optical source are set to 1 μs and 35 mW, respectively, and the delay time, TS, is taken as 0.1 μs. The spatial resolution for multi-points vibrations detection is ~110 m. The position error is less than ± 0.5 m when the number of averaging times is taken as 32. Without any optical amplifier, the sensing range is tested to be more than 16 km. Furthermore, other promotion technologies for the DOVFS based on Φ-OTDR, such as employing optical amplifiers to extend the sensing range, reducing the pulse width to improve spatial resolution and so on, are also compatible in our sensor. In comparison with the DOFVS based on the interference of continuous waves, our sensor has the capacity for measurement of simultaneous multi-points vibrations, and, in comparison with the DOFVS based on Φ-OTDR, the sensor has a simpler structure and a much lower requirement for internal device performance, which could reduce the cost dramatically.

References and links

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

2. R. Di Sante, “Fibre optic sensors for structural health monitoring of aircraft composite structures: recent advances and applications,” Sensors (Basel) 15(8), 18666–18713 (2015). [CrossRef]   [PubMed]  

3. P. R. Hoffman and M. G. Kuzyk, “Position determination of an acoustic burst along a Sagnac interferometer,” J. Lightwave Technol. 22(2), 494–498 (2004). [CrossRef]  

4. X. Fang, “Fiber-optic distributed sensing by a two-loop Sagnac interferometer,” Opt. Lett. 21(6), 444–446 (1996). [CrossRef]   [PubMed]  

5. S. J. Russell, K. R. C. Brady, and J. P. Dakin, “Real-time location of multiple time-varying strain disturbances, acting over a 40-km fiber section, using a novel dual-Sagnac interferometer,” J. Lightwave Technol. 19(2), 205–213 (2001). [CrossRef]  

6. S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15(6), 972–976 (1997). [CrossRef]  

7. A. A. Chtcherbakov, P. L. Swart, and S. J. Spammer, “Mach-Zehnder and Modified sagnac-distributed fiber-optic impact sensor,” Appl. Opt. 37(16), 3432–3437 (1998). [CrossRef]   [PubMed]  

8. S. Xie, Q. Zou, L. Wang, M. Zhang, Y. Li, and Y. Liao, “Positioning error prediction theory for dual Mach–Zehnder interferometric vibration sensor,” J. Lightwave Technol. 29(3), 362–368 (2011). [CrossRef]  

9. X. Hong, J. Wu, C. Zuo, F. Liu, H. Guo, and K. Xu, “Dual Michelson interferometers for distributed vibration detection,” Appl. Opt. 50(22), 4333–4338 (2011). [CrossRef]   [PubMed]  

10. 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]  

11. 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]  

12. Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol. 28(22), 3243–3249 (2010).

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

14. F. Peng, H. Wu, X.-H. Jia, Y.-J. Rao, Z.-N. Wang, and Z.-P. Peng, “Ultra-long high-sensitivity Φ-OTDR for high spatial resolution intrusion detection of pipelines,” Opt. Express 22(11), 13804–13810 (2014). [CrossRef]   [PubMed]  

15. C. Pan, H. Zhu, B. Yu, Z. Zhu, and X. Sun, “Distributed optical-fiber vibration sensing system based on differential detection of differential coherent-OTDR,” in Proceedings of IEEE Conference on Sensors (IEEE, 2012), 1–3. [CrossRef]  

16. C. Pan, H. Ye, M. Li, S. Zhao, and X. Sun, “Compensation method for blind segments of distributed optical-fiber vibration sensor based on differential-coherent OTDR,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.21. [CrossRef]  

17. B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17(1), R1–R16 (2006). [CrossRef]  

18. J. Beller, Fiber Optic Test and Measurement (Prentice Hall, 1998), Chap. 11.

References

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  1. X. Bao and L. Chen, “Recent progress in distributed fiber optic sensors,” Sensors (Basel) 12(12), 8601–8639 (2012).
    [Crossref] [PubMed]
  2. R. Di Sante, “Fibre optic sensors for structural health monitoring of aircraft composite structures: recent advances and applications,” Sensors (Basel) 15(8), 18666–18713 (2015).
    [Crossref] [PubMed]
  3. P. R. Hoffman and M. G. Kuzyk, “Position determination of an acoustic burst along a Sagnac interferometer,” J. Lightwave Technol. 22(2), 494–498 (2004).
    [Crossref]
  4. X. Fang, “Fiber-optic distributed sensing by a two-loop Sagnac interferometer,” Opt. Lett. 21(6), 444–446 (1996).
    [Crossref] [PubMed]
  5. S. J. Russell, K. R. C. Brady, and J. P. Dakin, “Real-time location of multiple time-varying strain disturbances, acting over a 40-km fiber section, using a novel dual-Sagnac interferometer,” J. Lightwave Technol. 19(2), 205–213 (2001).
    [Crossref]
  6. S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15(6), 972–976 (1997).
    [Crossref]
  7. A. A. Chtcherbakov, P. L. Swart, and S. J. Spammer, “Mach-Zehnder and Modified sagnac-distributed fiber-optic impact sensor,” Appl. Opt. 37(16), 3432–3437 (1998).
    [Crossref] [PubMed]
  8. S. Xie, Q. Zou, L. Wang, M. Zhang, Y. Li, and Y. Liao, “Positioning error prediction theory for dual Mach–Zehnder interferometric vibration sensor,” J. Lightwave Technol. 29(3), 362–368 (2011).
    [Crossref]
  9. X. Hong, J. Wu, C. Zuo, F. Liu, H. Guo, and K. Xu, “Dual Michelson interferometers for distributed vibration detection,” Appl. Opt. 50(22), 4333–4338 (2011).
    [Crossref] [PubMed]
  10. 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]
  11. 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]
  12. Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol. 28(22), 3243–3249 (2010).
  13. T. Zhu, Q. He, X. Xiao, and X. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013).
    [Crossref] [PubMed]
  14. F. Peng, H. Wu, X.-H. Jia, Y.-J. Rao, Z.-N. Wang, and Z.-P. Peng, “Ultra-long high-sensitivity Φ-OTDR for high spatial resolution intrusion detection of pipelines,” Opt. Express 22(11), 13804–13810 (2014).
    [Crossref] [PubMed]
  15. C. Pan, H. Zhu, B. Yu, Z. Zhu, and X. Sun, “Distributed optical-fiber vibration sensing system based on differential detection of differential coherent-OTDR,” in Proceedings of IEEE Conference on Sensors (IEEE, 2012), 1–3.
    [Crossref]
  16. C. Pan, H. Ye, M. Li, S. Zhao, and X. Sun, “Compensation method for blind segments of distributed optical-fiber vibration sensor based on differential-coherent OTDR,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.21.
    [Crossref]
  17. B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17(1), R1–R16 (2006).
    [Crossref]
  18. J. Beller, Fiber Optic Test and Measurement (Prentice Hall, 1998), Chap. 11.

2015 (1)

R. Di Sante, “Fibre optic sensors for structural health monitoring of aircraft composite structures: recent advances and applications,” Sensors (Basel) 15(8), 18666–18713 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (1)

2012 (1)

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

2011 (2)

2010 (1)

2007 (1)

2006 (1)

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17(1), R1–R16 (2006).
[Crossref]

2005 (1)

2004 (1)

2001 (1)

1998 (1)

1997 (1)

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15(6), 972–976 (1997).
[Crossref]

1996 (1)

Bao, X.

Brady, K. R. C.

Chen, L.

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

Y. Lu, T. Zhu, L. Chen, and X. Bao, “Distributed vibration sensor based on coherent detection of phase-OTDR,” J. Lightwave Technol. 28(22), 3243–3249 (2010).

Choi, K. N.

Chtcherbakov, A. A.

A. A. Chtcherbakov, P. L. Swart, and S. J. Spammer, “Mach-Zehnder and Modified sagnac-distributed fiber-optic impact sensor,” Appl. Opt. 37(16), 3432–3437 (1998).
[Crossref] [PubMed]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15(6), 972–976 (1997).
[Crossref]

Culshaw, B.

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17(1), R1–R16 (2006).
[Crossref]

Dakin, J. P.

Di Sante, R.

R. Di Sante, “Fibre optic sensors for structural health monitoring of aircraft composite structures: recent advances and applications,” Sensors (Basel) 15(8), 18666–18713 (2015).
[Crossref] [PubMed]

Fang, X.

Guo, H.

He, Q.

Hoffman, P. R.

Hong, X.

Jia, X.-H.

Juarez, J. C.

Kuzyk, M. G.

Li, Y.

Liao, Y.

Liu, F.

Lu, Y.

Maier, E. W.

Pan, C.

C. Pan, H. Zhu, B. Yu, Z. Zhu, and X. Sun, “Distributed optical-fiber vibration sensing system based on differential detection of differential coherent-OTDR,” in Proceedings of IEEE Conference on Sensors (IEEE, 2012), 1–3.
[Crossref]

Peng, F.

Peng, Z.-P.

Rao, Y.-J.

Russell, S. J.

Spammer, S. J.

A. A. Chtcherbakov, P. L. Swart, and S. J. Spammer, “Mach-Zehnder and Modified sagnac-distributed fiber-optic impact sensor,” Appl. Opt. 37(16), 3432–3437 (1998).
[Crossref] [PubMed]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15(6), 972–976 (1997).
[Crossref]

Sun, X.

C. Pan, H. Zhu, B. Yu, Z. Zhu, and X. Sun, “Distributed optical-fiber vibration sensing system based on differential detection of differential coherent-OTDR,” in Proceedings of IEEE Conference on Sensors (IEEE, 2012), 1–3.
[Crossref]

Swart, P. L.

A. A. Chtcherbakov, P. L. Swart, and S. J. Spammer, “Mach-Zehnder and Modified sagnac-distributed fiber-optic impact sensor,” Appl. Opt. 37(16), 3432–3437 (1998).
[Crossref] [PubMed]

S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac–Michelson interferometer for distributed disturbance detection,” J. Lightwave Technol. 15(6), 972–976 (1997).
[Crossref]

Taylor, H. F.

Wang, L.

Wang, Z.-N.

Wu, H.

Wu, J.

Xiao, X.

Xie, S.

Xu, K.

Yu, B.

C. Pan, H. Zhu, B. Yu, Z. Zhu, and X. Sun, “Distributed optical-fiber vibration sensing system based on differential detection of differential coherent-OTDR,” in Proceedings of IEEE Conference on Sensors (IEEE, 2012), 1–3.
[Crossref]

Zhang, M.

Zhu, H.

C. Pan, H. Zhu, B. Yu, Z. Zhu, and X. Sun, “Distributed optical-fiber vibration sensing system based on differential detection of differential coherent-OTDR,” in Proceedings of IEEE Conference on Sensors (IEEE, 2012), 1–3.
[Crossref]

Zhu, T.

Zhu, Z.

C. Pan, H. Zhu, B. Yu, Z. Zhu, and X. Sun, “Distributed optical-fiber vibration sensing system based on differential detection of differential coherent-OTDR,” in Proceedings of IEEE Conference on Sensors (IEEE, 2012), 1–3.
[Crossref]

Zou, Q.

Zuo, C.

Appl. Opt. (3)

J. Lightwave Technol. (6)

Meas. Sci. Technol. (1)

B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17(1), R1–R16 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Sensors (Basel) (2)

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

R. Di Sante, “Fibre optic sensors for structural health monitoring of aircraft composite structures: recent advances and applications,” Sensors (Basel) 15(8), 18666–18713 (2015).
[Crossref] [PubMed]

Other (3)

C. Pan, H. Zhu, B. Yu, Z. Zhu, and X. Sun, “Distributed optical-fiber vibration sensing system based on differential detection of differential coherent-OTDR,” in Proceedings of IEEE Conference on Sensors (IEEE, 2012), 1–3.
[Crossref]

C. Pan, H. Ye, M. Li, S. Zhao, and X. Sun, “Compensation method for blind segments of distributed optical-fiber vibration sensor based on differential-coherent OTDR,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2014), paper Th2A.21.
[Crossref]

J. Beller, Fiber Optic Test and Measurement (Prentice Hall, 1998), Chap. 11.

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Figures (12)

Fig. 1
Fig. 1 Scheme of the DOFVS based on the Sagnac interference in conjunction with the OTDR; SLD: super luminescent diode; C1, C2: 2 × 2 directional couplers; LT: delay fiber.
Fig. 2
Fig. 2 (a) Coherent OTDR traces with and without vibration detected at the output ports A and B (Values of the parameters are from the book [18]). (b) Coherent OTDR traces with 2-points vibrations detected at the output port A.
Fig. 3
Fig. 3 (a) Composite OTDR trace with 2-points vibrations. (b) Differential results of composite OTDR trace for different delay times.
Fig. 4
Fig. 4 Experimental setup for the DOFVS based on the Sagnac interferometer in conjunction with the OTDR; LS1: the first sensing fiber coil; LS2: the second sensing fiber coil; LS3: the third sensing fiber coil, PZT: piezo-electric transducer.
Fig. 5
Fig. 5 OTDR traces with and without vibration acting: (a) the OTDR traces detected at the output port A; (b) the OTDR traces detected at the output port B; (c) the output signals of the balanced detector (the composite OTDR traces).
Fig. 6
Fig. 6 Time-shifting differential results of the composite OTDR traces when vibration acting for different delay times, TS, of 0.5 μs, 1 μs, and 2 μs.
Fig. 7
Fig. 7 (a) Variations of amplitude of the vibration-induced pulse amplitude and the SNR with delay time, TS, for different optical input pulse widths of 1 μs and 2 μs. (b) Variations of the vibration-induced pulse width with delay time, TS, for different optical input pulse widths of 1 μs and 2 μs.
Fig. 8
Fig. 8 Detection of the vibration at ~6150 m: (a) 400 adjacent composite OTDR traces; (b) the time-shifting differential results of 400 adjacent composite OTDR traces (TS = 0.1 μs).
Fig. 9
Fig. 9 (a) Time response of the proposed DOFVS to vibration produced by the PZT driven by sine wave. (b) Frequency response of the sensor to vibration produced by the PZT driven by sine waves with different frequencies.
Fig. 10
Fig. 10 Detection of 2-points vibrations whose distance is 110 m: (a) composite OTDR traces with the abrupt increase and decrease at the second vibration position and their differential results; (b) differential results of 400 adjacent composite OTDR traces.
Fig. 11
Fig. 11 Variations of the detected vibration position with the observation time for different averaging times of 8, 16 and 32.
Fig. 12
Fig. 12 Detection of 2-points vibrations with the distance of ~16 km.

Equations (13)

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Δ t V = 2( l R l V ) v g ,
Δ t V (l) (l l V ) v g .
Δ t M = l T v g .
Δφ(l)={ 0 l< l V φ( l V v g )+φ[ l V v g +Δ t V (l)]φ( l V v g +Δ t M )φ[ l V v g +Δ t M +Δ t V (l)] l l V .
Δφ(l)Δφ( l V )=2[φ( l V v g )φ( l V v g +Δ t M )].
P AC (l)={ 1 4 A S P 0 [1exp(2αl)]exp(α l T ) l w 2 1 8 A S P 0 [2exp(αw)2]exp[α(2l+ l T )] w 2 <l l v , 1 8 A S P 0 {2exp(αw){{1cos[Δφ(l)]}exp[2α(l l V )] +{1+cos[Δφ(l)]}}}exp[α(2l+ l T )] l v <l l v + w 2 1 8 A S P 0 [exp(αw)1]{1+cos[Δφ(l)]}exp[α(2l+ l T )] l v + w 2 <l l S 1 8 A S P 0 {exp(aw)exp[2a(l l S )]}{1+cos[Δφ(l)]}exp[α(2l+ l T )] l S <l l S + w 2
A S =S α S α P 0 exp(αw)
P BC (l)={ 0 l l V 1 8 A S P 0 [exp(2α l V )exp(2αl)]{1cos[Δφ(l)]}exp(α l T ) l V <l l V + w 2 . 1 8 A S P 0 [exp(αw)1]{1cos[Δφ(l)]}exp[α(2l+ l T )] l V + w 2 <l l S 1 8 A S P 0 {exp(aw)exp[2a(l l S )]}{1cos[Δφ(l)]}exp[α(2l+ l T )] l S <l l S + w 2
P AC (l)={ 1 4 A S P 0 [1exp(2αl)]exp(α l T ) l w 2 1 8 A S P 0 [exp(aw)1]{1+cos[( j=0 i Δ φ j (l) )]} exp[a(2l+ l T )] w 2 <l l V1 (i=0) or l Vi + w 2 <l l Vi+1 (1i<N1) or l> l VN (i=N) 1 8 A S P 0 {{1+cos[ j=0 i1 Δ φ j (l) ]}exp(aw){{ cos[ j=0 i1 Δ φ j (l) ]cos[ j=0 i Δ φ j (l) ]}exp[2a(l l Vi )] +{1+cos[ j=0 i Δ φ j (l) ]}}}exp[a(2l+ l T )] l Vi <l l Vi + w 2 (1iN) 1 8 A S P 0 {exp(aw)exp[2a(l l S )]}{1+cos[( j=0 i Δ φ j (l) )]} exp[a(2l+ l T )] l S <l l S + w 2
P BC (l)={ 1 8 A S P 0 [exp(aw)1]{1 cos[ j=0 i Δ φ j (l) }exp[a(2l+ l T )] l l V1 (i=0) or l Vi + w 2 <l l Vi+1 (1i<N1) or l> l VN (i=N) 1 8 A S P 0 {{{1cos[ j=0 i1 Δ φ j (l) ]}exp(aw) {1cos[ j=0 i Δ φ j (l) ]}}exp(2al){cos[ j=0 i Δ φ j (l) ] cos[ j=0 i1 Δ φ j (l) ]}exp(2a l Vi )}exp(a l T ) l Vi <l l Vi + w 2 (1iN) 1 8 A S P 0 {exp(aw)exp[2a(l l S )]}{1 cos[ j=0 i Δ φ j (l) }exp[a(2l+ l T )] l S <l l S + w 2 ,
V(l)P(l)= k A P A-C (l) k B P B-C (l),
DV(l)=V(l+ T S v g 2 )V(l),
R S = T S v g +w 2 = v g 2 ( T S + T w ),

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