Evaluating and overcoming the impact of second echo in Brillouin echoes distributed sensing

Zhisheng Yang; Xiaobin Hong; Wenqiao Lin; Jian Wu

doi:10.1364/OE.24.001543

1. Introduction

Distributed optical fiber sensors based on Brillouin optical time-domain analysis (BOTDA) [1] have been widely employed for the monitoring of temperature and strain in large structures. In the conventional BOTDA technique, an optical pump pulse experiences energy transfer with a properly frequency-shifted counter-propagating continuous probe wave through stimulated Brillouin scattering (SBS) process. By scanning the frequency difference between these two waves, the Lorentzian Brillouin gain/loss spectrum (BGS/BLS) of each position along the fiber can be recovered. The center frequency of the BGS/BLS, where the maximum energy transfer occurs, represents the Brillouin frequency shift (BFS) that depends linearly on the fiber temperature and strain. Thus, any change of these quantities can be detected by fitting the measured BGS/BLS. The linewidth of the BGS/BLS, which corresponds to the convolution between the spectrum of the pump pulse and the natural Brillouin spectrum of the fiber [2], would get broadening as the width of the pump pulse decreases. This broadening associating with a reduced peak gain of BGS/BLS, degrades the performance of the BOTDA sensor in terms of BFS accuracy [3]. For this reason, a limit for good estimated BFS accuracy is given by the situation when pump pulse and Brillouin gain spectra show comparable bandwidth, corresponding to a pump pulse with a duration equal to a double phonon lifetime ( $2 \times τ_{A} \approx 12 n s$ ), or about 1 meter spatial resolution [2]. Numerous solutions have been developed to overcome this trade-off between the spatial resolution and the frequency accuracy, which can be classified into two types. One type performs the SBS process in optical correlation domain to achieve random-access sensing with very high spatial resolution, based on synchronously frequency modulation of the constant-amplitude pump and probe waves driven by a sine wave [4, 5], or on phase modulation of both driven by a pseudo-random bit sequence (PRBS) for extending the sensing range [6]. By combining the latter with a time gated envelope on pump [7–9], or even coding the time gate [10, 11], the sensing performance in terms of the signal-to-noise ratio (SNR) and the acquisition time can be greatly improved, leading to a largest number of interrogated points so far (8.8 km range and 2 cm resolution) with 211 scans of peak positions [11]. Although the acquisition time of the state of the art is significant reduced with respect to the BOCDA based techniques at early stage, it’s still two orders of magnitude slower than that of the conventional BOTDA.

Therefore, if the normal measurement time and the simplicity of the equipment have to be concerned, the other type of sub-meter spatial resolution sensors, which is more similar with the standard BOTDA, could be better considered. In this type of sensors, in addition to the single pulse that acts as interrogating signal, another part of optical wave is used in pump wave for electrostrictively pre-activating the acoustic wave through the Brillouin interaction with continuous probe waves. This additional part can be a short pulse that works for double-pulse BOTDA [12, 13], or be continuous wave in Brillouin echoes distributed sensing (BEDS) [14–18]. The latter can provide better SNR since the acoustic wave is sufficiently established due to the long pre-pumping part, but which suffers from following two penalties [18, 19]: i) the spontaneous Brillouin noise and strong pump depletion due to the continuous Brillouin interaction along the full length of the fiber, which limit the sensing range in km scale; ii) the second echo phenomenon that originates from the inertial feature of the acoustic wave, deteriorating the BFS estimation. When the km-scale sensing range is required, the problem imposed by second echo becomes more crucial. The deconvolution method [19] and the Brillouin gain-profile tracing [20] can mitigate the second echo, each of them needs post-processing algorithm. The differential pulse pair methods [18, 21] are capable of alleviating both limitations, with the compromise of one more acquisition process.

Although the deep analysis from literatures [18, 19] has proved that the second echo manifests as an unwanted contribution of the non-local response on the correct signal, which is decisively detrimental for the measurement, it still remains unclear that how severe the error on the BFS determination is, at a given situation, and what is the worst case scenario. In this paper, we try to answer above two questions by further developing the analytical solution of the Brillouin gain on the probe wave proposed in literature [19]. Our investigation shows that for a given spatial resolution, the second echo phenomenon has different behavior as the length of the heated/stressed section changes. The second echo has no significant negative impact when the fiber is uniform, while having most severe impact when the length of the heated/stressed fiber section is exactly equal to the spatial resolution. Under this worst condition, the systematic error on the estimated BFS is also mathematically quantified, showing a maximum error of 8.5 MHz when 17 MHz BFS change is applied on the heated/stressed section. A novel parabolic-amplitude four-section pulse based BEDS technique is proposed, which can compensate the second echo optically without using extra measurement time and post-processing algorithm. Moreover, an upgraded mathematic model is also provided, which can be used for optimizing the shape and the key parameters of the proposed four-section pulse by combining with the iterative algorithm. The experimental results validate the theoretical analysis, and demonstrate that the proposed method can almost entirely eliminate the impact of second echo.

2. Evaluating the impact of second echo in BEDS methods

In the BEDS methods, a continuous optical wave at pump frequency is used for electrostrictively activating the steady-state acoustic wave over all the fiber length through the Brillouin interaction with the continuous wave at probe frequency [19]. This pre-existing acoustic wave changes much more slowly than the two optical waves, showing an inertial feature and requiring a typical time equal to a multiple of its lifetime ( $τ_{A} \approx 6 n s$ ) to adapt to a new situation [15]. So if an abrupt change of either amplitude or phase is implemented on the continuous pump wave during a time period T being much shorter than 6 ns, the acoustic wave will not have notable change in phase and amplitude. Therefore, the interrogating pump pulse will be reflected unmodified in phase and amplitude and then be superposed on the probe wave during its T time period. This way the distributed sensing with high spatial resolution $T \times V_{g} / 2$ can be realized, meanwhile providing the natural BGS linewidth.

Because of the existing of the short abrupt change on the pump, generally the continuous pump wave consists of three sections [19]: $α$ section, $β$ section and $γ$ section, as shown in Fig. 1, where the sections $α$ and $γ$ represent the original continuous pump wave, the amplitude of both are set as 1 hereafter in all cases for the convenience; while the $β$ section is located between section $α$ and $γ$ , representing the real interrogating pulse (abrupt change implemented on the continuous pump wave), which can be amplitude- or phase-coded in several ways.

Fig. 1 The illustration of the concept of three-section pulse in BEDS methods.

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Typically, the $β$ section can be intensity-modulated as a bright pulse (Fig. 2(a)) [14–16] or a dark pulse (Fig. 2(b)) [17], or be phase-modulated as a $π$ phase shift pulse to obtain double response with respect to that of dark pulse method (Fig. 2(c)) [18], in which the amplitude of $β$ section is equivalent to −1.

Fig. 2 Three typical pulse configurations in BEDS methods. (a): Bright pulse; (b): Dark pulse; (c) $π$ phase shift pulse.

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One common problem existing in all three techniques is the second echo phenomenon, which has been discovered as one of the most detrimental effect in BEDS methods for km-range sensing. The origin of the second echo phenomenon can be interpreted as follows: when the $β$ section of the pulse passes through a given position of the sensing fiber, the steady-state acoustic field pre-excited by long $α$ section would experience a small change (decaying in dark pulse and $π$ phase shift pulse cases, while increasing in bright pulse case) during its T time period. After that, when the $γ$ section crosses the same position, the slightly modified acoustic field couldn’t fully match the counter-propagating pump and probe waves, and therefore, contributes differently (contributes less in dark pulse and π phase shift pulse cases, while contributes more in bright pulse case) with respect to the former steady-state acoustic field exited by $α$ section until it gradually turns back to its original level after a duration given by a multiple of the acoustic lifetime. In order to thoroughly evaluate this phenomenon, the mathematical model provided by [19] is employed here to bridge our theory that will be presented later. The model starts from solving the traditional instantaneous three-wave coupled equations:

\frac{\partial A_{p} (z, t)}{\partial z} + \frac{1}{V_{g}} \frac{\partial A_{p} (z, t)}{\partial t} = i \frac{1}{2} g_{2} A_{s} (z, t) Q (z, t),

\frac{\partial A_{s} (z, t)}{\partial z} - \frac{1}{V_{g}} \frac{\partial A_{s} (z, t)}{\partial t} = - i \frac{1}{2} g_{2} A_{p} (z, t) Q^{*} (z, t),

\frac{\partial Q (z, t)}{\partial t} + Γ_{A} Q (z, t) = i g_{1} A_{p} (z, t) A_{s}^{*} (z, t),

where

A_{p} (z, t)

,

A_{s} (z, t)

and

Q (z, t)

are slow varying complex envelopes of pump wave, probe wave and acoustic wave, respectively;

Γ_{A}

is the frequency detuning parameter, which equals to

i (Ω_{B}^{2} - Ω^{2} - i Ω Γ_{B}) / 2 Ω

, where

Γ_{B}

is the acoustic damping constant that is an important parameter to determine the full width of half maximum (FWHM) of the Brillouin gain spectrum

Δ ν_{B}

, as

Γ_{B} = 1 / τ_{A} = 2 π Δ ν_{B}

;

Ω_{B} / 2 π

and

Ω / 2 π

are Brillouin resonance frequency and pump-probe frequency difference, respectively;

g_{1}

and

g_{2}

are electrostrictive and elasto-optic coupling coefficients, respectively.

By solving Eq. (1-3) and substituting the boundary conditions of the three typical BEDS methods (i.e., $α = γ = 1$ , $β =$ 2, 0 or −1) into the analytical solution, the variations of probe wave observed at z = 0 as a function of time, which is attributed to a very short section $Δ z$ at one given position $z_{N}$ , can be given by:

\begin{array}{l} a_{s}^{b r i g h t} (z = 0, z_{N}, t) = \frac{g (z_{N}) I_{p}^{0} A_{s}^{0} Δ z}{2 Γ_{A}^{*}} \times \\ {u (t_{1}) - u (t_{2}) + [4 - 2 \exp (- Γ_{A}^{*} t_{2})] [u (t_{2}) - u (t_{3})] + [1 - [1 - \exp (Γ_{A}^{*} T)] \exp (- Γ_{A}^{*} t_{2})] u (t_{3})}, \end{array}

\begin{array}{l} a_{s}^{d a r k} (z = 0, z_{N}, t) = \frac{g (z_{N}) I_{p}^{0} A_{s}^{0} Δ z}{2 Γ_{A}^{*}} \times \\ {u (t_{1}) - u (t_{2}) + [1 + [1 - \exp (Γ_{A}^{*} T)] \exp (- Γ_{A}^{*} t_{2})] u (t_{3})}, \end{array}

\begin{array}{l} a_{s}^{π} (z = 0, z_{N}, t) = \frac{g (z_{N}) I_{p}^{0} A_{s}^{0} Δ z}{2 Γ_{A}^{*}} \times \\ {u (t_{1}) - u (t_{2}) + [1 - 2 \exp (- Γ_{A}^{*} t_{2})] [u (t_{2}) - u (t_{3})] + [1 + 2 [1 - \exp (Γ_{A}^{*} T)] \exp (- Γ_{A}^{*} t_{2})] u (t_{3})}, \end{array}

where

t_{1} = t - 2 z_{N} / V_{g}

,

t_{2} = t - t_{0} - 2 z_{N} / V_{g}

and

t_{3} = t - t_{0} - T - 2 z_{N} / V_{g}

;

u (\cdot)

represents the Heaviside unit step function. In Eq. (4-6), the terms corresponding to the time region

[u (t_{2}) - u (t_{3})]

denote the correct response induced by interrogating pulse, while the terms corresponding to the time region

u (t_{3})

denote the unwanted response imposed by the second echo. Figure 3(a) illustrates the calculated waveforms expressed by Eq. (4-6) at the Brillouin resonance frequency of position

z_{N}

, respectively, by assuming T = 5 ns (corresponding to 50 cm spatial resolution), which clearly shows that the second echo manifests as a long exponential tail (part B) right after the correct response (part A) in each waveform.

Fig. 3 (a):The Brillouin temporal waveforms at position $z_{N}$ obtained at the Brillouin resonance frequency; (b) The illustration of the convolution between the Brillouin waveforms and the fiber impulse response (assuming the fiber is uniform).

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It should be noted that the $a_{s}^{} (z = 0, z_{N}, t)$ is the temporal waveform of the Brillouin gain at a given position $z_{N}$ , which is not the BOTDA trace that we normally acquired from the oscilloscope. Since every fiber, spanning the range from $z_{0}$ to $z_{L}$ , can be considered as a concatenation of M very short segments ( $M = (z_{L} - z_{0}) / Δ z$ ), the BOTDA trace, which is actually proportional to the sum of the contributions of individual segments, can be written as:

a_{s} (z = 0, t) = \sum_{N = 0}^{L} a_{s} (z = 0, z_{N}, t),

as shown in Fig. 3(b). It can be seen that the contributions from the unwanted long tails belonging to their corresponding positions (

z_{N - 1}

,

z_{N - 2}

,…, and so on) affect the signal that should be only from position

z_{N}

after the summation process. It means that the long tails imposed by the second echo carry the Brillouin gain/loss information from their corresponding fiber positions (extends over a few meters right after the correct interrogated segment), contaminating the local spatial information that should only be attributed to the interrogating pulse. This detrimental impact shows different behaviors in different situations. One extreme situation is that the BFS of the interrogating segment and its neighboring segment covered by the second echo are the same (i.e., the fiber is uniform, or the length of a hot spot is longer than a few meters), this way the BGS attributed by the long tails overlaps the correct BGS attributed by sensing pulse, making the impact of second echo indiscernible. The other extreme situation is that a hot spot having the same length as spatial resolution is interrogated, thus the BGS attributed by the second echo is completely from the positions outside of the hot spot, imposing a maximum unwanted BGS on the correct BGS. According to Eq. (7) and Fig. 3(b), in this worst case, the peak amplitudes of both BGS can be obtained by integrating the part A and part B in Fig. 3(a) over their corresponding time regions (from

t_{2}

to

t_{3}

for the correct signal, and from

t_{3}

to

\infty

for the unwanted signal imposed by second echo), respectively:

\frac{A_{co r r e c t}^{b r i g h t}}{A_{u n w a n t e d}^{b r i g h t}} = \frac{\int_{t_{2}}^{t_{3}} [a_{s}^{b r i g h t} (0, z_{N}, t) - 1] d t}{\int_{t_{3}}^{\infty} [a_{s}^{b r i g h t} (0, z_{N}, t) - 1] d t} = \frac{[3 T + \frac{2}{Γ_{A}^{*}} (\exp [- Γ_{A}^{*} T] - 1)]}{[\frac{1}{Γ_{A}^{*}} (1 - \exp [- Γ_{A}^{*} T])]} \approx \frac{T}{T},

\frac{A_{c o r r e c t}^{d a r k}}{A_{u n w a n t e d}^{d a r k}} = \frac{\int_{t_{2}}^{t_{3}} [a_{s}^{d a r k} (0, z_{N}, t) - 1] d t}{\int_{t_{3}}^{\infty} [a_{s}^{d a r k} (0, z_{N}, t) - 1] d t} = \frac{- T}{[\frac{1}{Γ_{A}^{*}} (\exp [- Γ_{A}^{*} T] - 1)]} \approx \frac{- T}{- T},

\frac{A_{c o r r e c t}^{π}}{A_{u n w a n t e d}^{π}} = \frac{\int_{t_{2}}^{t_{3}} [a_{s}^{π} (0, z_{N}, t) - 1] d t}{\int_{t_{3}}^{\infty} [a_{s}^{π} (0, z_{N}, t) - 1] d t} = \frac{[\frac{2}{Γ_{A}^{*}} (\exp [- Γ_{A}^{*} T] - 1)]}{[\frac{2}{Γ_{A}^{*}} (\exp [- Γ_{A}^{*} T] - 1)]} \approx \frac{- 2 T}{- 2 T} .

During the mathematic deducing, all the terms $(\exp [- Γ_{A}^{*} T] - 1)$ are approximated as $- Γ_{A}^{*} T$ , since the term $- Γ_{A}^{*} T$ is small enough to be close to 0. From the results in Eq. (8-10), it can be seen that the peak amplitude of the BGS attributed to the second echo is completely equal to that attributed to the sensing pulse in any BEDS method, and these two amplitudes are always identical. Moreover, this equal-amplitude feature is independent of the value of parameter T (pulse width that corresponds to the spatial resolution), as long as T is less than 12 ns acoustic decaying time (corresponding to 1.2 m spatial resolution). Therefore, in spectral domain, the mathematic expression of the synthetic BGS for all three BEDS methods can be simply written as the sum of two equal-amplitude but center-shifted Lorentz spectra:

g_{B E D S} (Ω) = \frac{{(Γ_{B} / 2)}^{2}}{{(Ω - Ω_{B})}^{2} + {(Γ_{B} / 2)}^{2}} + \frac{{(Γ_{B} / 2)}^{2}}{{[Ω - (Ω_{B} + Δ Ω_{B})]}^{2} + {(Γ_{B} / 2)}^{2}},

where

Ω_{B} / 2 π

denotes the BFS of the interrogated segment (we call it hot spot hereafter), and

(Ω_{B} + Δ Ω_{B}) / 2 π

denotes the BFS of the segment covered by the second echo (we call it unheated segment hereafter), which is neighboring of the interrogated segment. According to Eq. (11), Figs. 4(a)-4(d) exemplify the simulated BGS in four cases of Brillouin frequency differences (5 MHz, 15 MHz, 25 MHz and 35 MHz) between the unheated segment and the hot spot by assuming the FWHM of the natural BGS is 30 MHz, in which the red lines stand for the synthetic BGS inside the hot spot affected by second echo, while the black lines and blue lines that stand for the correct BGS of the hot spot and that of the neighboring unheated segment, respectively, are plotted as references. It can be seen that when the BFS difference applied on the hot spot is relative small (corresponding to the cases of Figs. 4(a) and 4(b)), the synthetic BGS has only one peak located exactly at the center between the two resonance frequencies, which would directly lead to the system error on BFS estimation. As the BFS difference becomes larger (corresponding to the cases of Figs. 4(c) and 4(d)), the synthetic BGS has two peaks with equal amplitude, inducing the ambiguity in the BFS recognizing process, which may be overcome by using advanced fitting algorithm.

Fig. 4 The simulated BGS in cases of (a): 5 MHz; (b): 15 MHz; (c): 25 MHz; (d): 35 MHz BFS differences between the hot spot and the neighboring unheated segment.

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In order to find the boundary condition of these two situations, the derivation operation is applied to Eq. (11) to look for the extreme points of the synthetic spectrum:

g^{'}_{B E D S} (Ω) = \frac{- 2 Γ_{B}^{2} (Ω - Ω_{B})}{{[4 {(Ω - Ω_{B})}^{2} + Γ_{B}^{2}]}^{2}} + \frac{- 2 Γ_{B}^{2} (Ω - Ω_{B} - Δ Ω_{B}^{})}{{[4 {(Ω - Ω_{B} - Δ Ω_{B}^{})}^{2} + Γ_{B}^{2}]}^{2}} = 0,

which can be reformulated as:

\begin{array}{l} 2 (^{Ω - Ω_{B}) 5} - 5 Δ Ω_{B}^{} (^{Ω - Ω_{B}) 4} + (6 Δ Ω_{B}^{2} + Γ_{B}^{2}) (^{Ω - Ω_{B}) 3} - (6 Δ Ω_{B}^{3} + \frac{3}{2} Γ_{B}^{2} Δ Ω_{B}^{}) (^{Ω - Ω_{B}) 2} \\ + (Δ Ω_{B}^{4} + \frac{1}{2} Γ_{B}^{2} Δ Ω_{B}^{2} + \frac{1}{8} Γ_{B}^{4}) (Ω - Ω_{B}) - \frac{1}{16} Γ_{B}^{4} Δ Ω_{B}^{} = 0. \end{array}

Since Eq. (13) is a five-degree polynomial equation, there should be five solutions that can be solved. However, only three of them are valid solutions (i.e., real solutions):

Ω_{i} = Ω_{B} + \frac{1}{2} (Δ Ω_{B} \pm \sqrt{- Δ Ω_{B}^{2} - Γ_{B}^{2} + 2 \sqrt{Δ Ω_{B}^{2} (Δ Ω_{B}^{2} + Γ_{B}^{2})}}), i = 1, 2,

Ω_{3} = Ω_{B} + \frac{1}{2} Δ Ω_{B},

provided that the relationship

Δ Ω_{B}^{2} + Γ_{B}^{2} < 2 \sqrt{Δ Ω_{B}^{2} (Δ Ω_{B}^{2} + Γ_{B}^{2})}

can be satisfied (i.e.,

Δ Ω_{B} > \frac{\sqrt{3}}{3} Γ_{B}

or

Δ Ω_{B} < - \frac{\sqrt{3}}{3} Γ_{B}

), meaning that the BFS difference between the hot spot and the unheated segment must be more than about 17 MHz (assuming

Γ_{B} = 2 π \times 30 M H z

). In such a double-peak BGS case, although some advanced fitting algorithms may be capable of selecting the peak belonging to the hot spot (the peak on the right hand of the red curve in Figs. 4(c) and 4(d)), there is still slightly difference between this spectral location and the correct BFS, since the superposition of two BGS modifies the peak locations mutually. On the contrary, when the condition becomes

Δ Ω_{B}^{2} + Γ_{B}^{2} \geq 2 \sqrt{Δ Ω_{B}^{2} (Δ Ω_{B}^{2} + Γ_{B}^{2})}

(i.e.,

- \frac{\sqrt{3}}{3} Γ_{B} \leq Δ Ω_{B} \leq \frac{\sqrt{3}}{3} Γ_{B}

), corresponding to the situation that the BFS difference is less than 17 MHz, there is only one valid solution:

Ω_{1} = Ω_{B} + \frac{1}{2} Δ Ω_{B},

which indicates that the BGS has only one peak. In this case, apparently the BFS estimation error is

0.5 Δ Ω_{B} / 2 π

that depends on the original BFS difference between the hot spot and the unheated segment, which can reach up to 8.5 MHz when 17 MHz BFS change occurs.

In order to intuitively show all the frequency errors discussed above, Fig. 5 illustrates the estimated BFS (blue curve) obtained from the synthetic BGS versus different BFS applied on the hot spot, by assuming the BFS of the unheated segment is 10.74 GHz. Additionally, the correct BFS relationship (green dashed curve) is also provided as a reference. The red curve depicts the trend of frequency estimation error as a function of BFS in hot spot, showing that the largest error occurs at the boundary frequency (17 MHz) between single-peak BGS and double-peak BGS situations.

Fig. 5 Blue curve: the estimated BFS of the hot spot in BEDS methods versus the BFS applied on the hot spot; Green dashed curve: the correct BFS versus the BFS applied on the hot spot; Red curve: the BFS error versus the BFS applied on the hot spot.

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3. Proposed four-section pump pulse based BEDS method

Aiming at suppressing the impact of second echo as much as possible, the BEDS methods based on a novel four-section pump pulse is proposed in this paper. Being different from the traditional three-section pump pulse discussed in section 2, an additional β′section with slightly different amplitude is deployed between β section and γ section in this new pulse configuration, which is illustrated as the red section in Fig. 6. Since we have proved that the impact of the second echo phenomenon is identical in all these three BEDS methods, hereafter we just employ the dark pulse based scheme as an example to carry out all analysis.

Fig. 6 The proposed four-section pulse configurations for BEDS methods. (a): Bright pulse; (b): Dark pulse; (c) $π$ phase shift pulse.

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In the four-section dark pulse based technique, the pre-excited steady-state acoustic field at a given position of the fiber is also established by the Brillouin interaction between the probe wave and the long $α$ section in the pump, and then also experiences decaying during the $β$ section with its short duration T. However, after that, the slightly decayed acoustic wave experiences Brillouin interaction with the probe signal and the $β^{'}$ section having slightly higher amplitude than that of continuous wave (i.e., the amplitude of $α$ and $γ$ sections) during its duration $T^{'}$ . In this period, the added contribution on the probe signal attributed to the added amplitude of the $β^{'}$ section can be expected to fully compensate the reduced contribution from the decayed acoustic wave, if the amplitude ( $β^{'}$ ) and the duration ( $T^{'}$ ) are properly chosen. In other words, these two parameters must be chosen to ensure that the unwanted long exponential tail on the Brillouin gain waveform (part B in Fig. 3) could be completed removed. Mathematically, this condition for optimizing the $β^{'}$ and the $T^{'}$ can be expressed as:

α \times Q (z_{0}, t_{0}) = {\begin{cases} β^{'} (t_{0} + t) \times Q (z_{0}, t_{0} + t), T \leq t \leq T + T^{'} \\ γ \times Q (z_{0}, t_{0} + t), t > T + T^{'} \end{cases}

which means that the amplitude of the part B in Fig. 3 must be flat (i.e., equal to 1 all the time). It should be noted that the amplitude of

β^{'}

section as a function of time can be arbitrarily designed, the constant-amplitude red square in Fig. 6 is just illustrated as a primary concept. Obviously, the Eq. (17) cannot be satisfied if the amplitude of the

β^{'}

section keeps constant from

t_{0} + T

to

t_{0} + T + T^{'}

, since the amplitude of acoustic wave, which gradually turns back to its steady-state level exponentially during this period, is not constant. In order to adapt the rising trend of the acoustic wave, the declining amplitude of

β^{'}

section must follow, for instance, a parabolic distribution, as shown in Fig. 7.

Fig. 7 The configuration of the four-section dark pulse with parabolic-amplitude $β^{'}$ section.

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The mathematic form of this four-section dark pulse with parabolic-amplitude $β^{'}$ section can be expressed as:

\begin{array}{l} A_{P} (z, t) = u (t - z / v_{g}) - u (t - t_{0} - z / v_{g}) \\ + [\frac{c - 1}{{T^{'}}^{2}} {(t - t_{0} - T - T^{'} - z / v_{g})}^{2} + 1] \times [u (t - t_{0} - T - z / v_{g}) - u (t - t_{0} - T - T^{'} - z / v_{g})] \\ + u (t - t_{0} - T - T^{'} - z / v_{g}) . \end{array}

By substituting Eq. (18) into Eq. (1-3), the analytical solution for the variations of probe wave observed at z = 0 as a function of time, which is attributed to a very short section $Δ z$ at one given position $z_{N}$ , can be resolved as:

\begin{array}{l} a_{s} (0, z_{N}, t) = \frac{g (z_{N}) I_{p}^{0} A_{s}^{0} Δ z}{2 Γ_{A}^{*}} \times {\begin{cases} u (t_{1}) - u (t_{2}) + [1+ \exp (- Γ_{A}^{*} t_{2}) - \exp (- Γ_{A}^{*} t_{3})] [(k + 1) u (t_{3}) - k u (t_{4})] \end{cases} \\ + \frac{c - 1}{{T^{'}}^{2}} {t_{3}^{2} - \frac{2 + 2 T^{'} Γ_{A}^{*}}{Γ_{A}^{*}} \times t_{3} + (\frac{2 + 2 T^{'} Γ_{A}^{*}}{Γ_{A}^{*}^{2}} + {T^{'}}^{2}) [1 - \exp (- Γ_{A}^{*} t_{3})]} [(k + 1) u (t_{3}) - k u (t_{4})] \\ \begin{array}{l} + \frac{1 - c}{{T^{'}}^{2}} {t_{4}^{2} - \frac{2}{Γ_{A}^{*}} \times t_{4} + \frac{2}{Γ_{A}^{*}^{2}} [1 - \exp (- Γ_{A}^{*} t_{4})]} u (t_{4}) \end{array}}, \end{array}

where

k = (c - 1) t_{4}^{2} / {T^{'}}^{2}

;

t_{1} = t - 2 z_{N} / V_{g}

,

t_{2} = t - t_{0} - 2 z_{N} / V_{g}

,

t_{3} = t - t_{0} - T - 2 z_{N} / V_{g}

and

t_{4} = t - t_{0} - T - T^{'} - 2 z_{N} / V_{g}

. In Eq. (19), there are two unknown parameters

c

and

T^{'}

, which represent the initial amplitude and the width of

β^{'}

section, respectively. It’s worth mentioning that the parabolic-amplitude

β^{'}

section cannot perfectly satisfy the optimal condition expressed by Eq. (17) all the time, since the declining trend of parabolic function cannot perfectly match the rising trend of the recovery acoustic wave. Therefore, the iterative algorithm is employed for optimizing those two parameters by combining Eqs. (17) and (19), ensuring that the amplitude distribution of the waveform (expressed by Eq. (19)) after

t_{3}

could be as close to 1 as possible.

For instance, for the 1 ns pulse width (10 cm spatial resolution) that will be used in the experimental part of this paper, the corresponding optimal value of the parameters can be solved as $c = 1.09$ and $T^{'} = 1 8 n s$ . Figure 8(b) illustrates the simulation of the Brillouin temporal waveform (similar with the waveform in Fig. 3) by using parabolic-amplitude four-section dark pulse (the red curve) and the traditional three-section dark pulse (the blue curve), which clearly shows that the maximum difference between 1 and the amplitude of the red curve is less than 0.1%, meaning that the unwanted exponential tail existing in the proposed technique is almost suppressed. In order to highlight that the parabolic-amplitude $β^{'}$ section has relative better performance, the Brillouin gain in cases of constant-amplitude and linear-amplitude $β^{'}$ section are also calculated by using same parameters for comparison, as the black curve and green curve shown in Fig. 8(b), respectively. Although those two schemes can also compensate the second echo precisely at the next point of the sensing region (at 22 ns), after that the overcompensation appears immediately.

Fig. 8 (a): The configuration of the four-section dark pulse with different shapes of $β^{'}$ section; (b): The Brillouin gain waveforms at the position of $z_{0}$ with the Brillouin resonance frequency by using four-section dark pulse with different shapes of $β^{'}$ section; Inset: zoom of the part with maximum difference between 1 and the amplitude of red curve.

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3. Experimental results and discussion

The experimental setup that has been implemented to validate the proposed method is shown in Fig. 9. A narrow linewidth (15 KHz) tunable laser source at 1550 nm with 18dBm optical power is split into pump and probe waves by a 50:50 coupler. In the probe branch (upper arm in the figure), the continuous-wave light is injected into a high extinction ratio (40 dB) electro-optic modulator (EOM1), operating at the null transmission point to generate a carrier-suppressed double-sideband (SC-DSB) wave, using a varying RF modulating frequency being step-swept by 1 MHz over the BGS of the sensing fiber. A polarization switch (PS) is then used to minimize the impact of the polarization-dependent Brillouin gain, and an optical isolator is used to protect the laser and other optical components. In the pump branch, the optical four-section pulse is generated by intensity modulation using EOM2, driven by the electrical signal that has been programmed from an arbitrary waveform generator (AWG) with 3 GHz bandwidth and 10 G/s sampling rate. By adjusting the EDFA and the tunable attenuator after the EOM2, the pump power can be properly set. In the receiver part, one sideband of the probe wave at lower frequency (BOTDA trace of Brillouin gain) is detected by a photo-detector with bandwidth of 1 GHz, and finally received by an oscilloscope with 1 GHz bandwidth and 10 G/s sampling rate, with 512 average times per trace.

Fig. 9 Experimental setup of the proposed method. EOM: electro-optic modulator. AWG: arbitrary waveform generator; PC: polarization controller; PS: polarization switch; TA: tunable attenuator; EDFA: erbium-doped fiber amplifier, FBG: fiber Bragg grating; PD: photodetector.

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It’s worth mentioning that the proposed four-section dark pulse method focuses on mitigating the BFS estimation error induced by the second echo phenomenon, while the other limitation in terms of sensing length existing in traditional dark pulse method, which is imposed by the strong spontaneous Brillouin noise from the long pump pulse, cannot be overcome, since the proposal just places an additional short section inside the original continuous pump wave. Therefore, there is neither improvement nor deterioration on sensing length by using the new four-section pulse. In order to address the improvement in terms of the annihilation of the second echo, a short sensing fiber with the length of 10.3 m is deliberately used to obtain relative clear results, at the middle of which a short stressed fiber section with different lengths and different BFS are applied.

In order to evaluate the improvement of the proposal, the method based on the traditional three-section dark pulse and the proposed four-section dark pulse are both carried out in our experimental demonstration. The width of the sensing pulse for both methods is set as 1 ns, corresponding to 10 cm spatial resolution. The parameters of the optical parabolic-amplitude four-section pulse are set as follows: $p_{α} = p_{γ} = 60 m w$ , $p_{c} = 71 m w$ (being ${1.09}^{2}$ times higher than the power of $α$ and $γ$ section), and $T^{'} = 1 8 n s$ according to the theoretical analysis in section 2. At the middle of sensing fiber (5.6 m), a section with the length exactly equal to the spatial resolution is applied by changing the stress on it, making its BFS has 30 MHz frequency shift (10.762 GHz) with respect to the uniform part of the sensing fiber (10.732 GHz).

As a first impression, the top view of 3D-mappings of Brillouin gain distribution versus location and scanning frequency by using both techniques are shown in Figs. 10(a) and 10(b), respectively. For the sake of visual clarity, we only show the Brillouin gain distribution from 5.2 m to 6.2 m. Although the stressed section could be observed in Fig. 10(a), the color contrast at that location (at 5.6 m) is relative poor, implying that the Brillouin gain there at 10.732 GHz is similar with the one at 10.762 GHz. However, by using the proposed four-section dark pulse, the stressed section can be clearly seen with a good color contrast. Besides, the Brillouin gain over the uniform part of the fiber by using the three-section dark pulse shows better SNR than that of four-section dark pulse, indicating that the Brillouin gain for three-section dark pulse is reinforced by additional contribution, which will be clarified hereafter.

Fig. 10 The Brillouin gain distribution versus position and scanning frequency by using (a) three-section dark pulse and (b) proposed four-section dark pulse.

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For observing the improvement clearly, the Brillouin gain distribution versus position at two typical scanning frequencies (10.732 GHz and 10.762 GHz) by using both methods are shown in Fig. 11(a) and 11(b), respectively. Ideally, the Brillouin gain at the stressed section should reach maximum at its corresponding BFS (i.e., the scanning frequency is 10.762 GHz), while being much lower at the scanning frequency of 10.732GHz because of the 30 MHz frequency difference. However, when the three-section dark pulse is used, the above mentioned Brillouin gains (points A and B shown in Fig. 11(a)) at those two scanning frequencies have very similar amplitude, indicating that there will be two identical peaks located at 10.732 GHz and 10.762 GHz in its BGS, respectively. The reason comes from the fact that the second echo imposes an identical contribution with shifted spectral position on the correct signal, as discussed previously. Moreover, from Fig. 11(a) it can be seen that in the uniform part of the fiber, this unwanted contribution that has the same spectral position as the correct signal, magnifies the amplitude difference between the red and blue curves, resulting in relative better SNR. By using the proposed four-section dark pulse method, the Brillouin curves are correctly obtained, in which the peak point of the Brillouin gain at 10.762 GHz has much higher amplitude than the one at 10.732 GHz at the stressed section, and the amplitude difference between them over the uniform part of the fiber is correct, as shown in Fig. 11(b).

Fig. 11 The Brillouin gain traces versus position at the scanning frequency of 10.732 GHz and 10.762 GHz by using (a) three-section dark pulse and (b) proposed four-section dark pulse.

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For more intuitive demonstration, using the same measured data as in Fig. 11 but being plotted in spectral domain, the BGS at the stressed fiber section (5.6 m) and the one at uniform part (5 m) of the fiber are shown in Fig. 12(a). The double peaks in the BGS at 5.6m and the magnified amplitude of the BGS at 5 m can be clearly observed using the three-section dark pulse, while all the spectra can be correctly obtained by using the proposal. Furthermore, both methods are realized by using 2 ns pulse width, respectively, as shown in Fig. 12(b). The parameters of the proposed four-section pulse are re-optimized as follows: $p_{α} = p_{γ} = 60 m w$ , $p_{c} = 83.5 m w$ (being ${1.18}^{2}$ times higher than the power of $α$ and $γ$ section), and $T^{'} = 1 7 n s$ accordingly, by simply changing the electrical signal from the AWG. Note that the length of the stressed section of the fiber is also changed to 20 cm for maintaining the worst condition (the length of the interrogated segment is exactly equal to the spatial resolution). In Fig. 12(b), the Brillouin gain spectra being similar with those in Fig. 12(a) can be observed, demonstrating the point that the contribution of the second echo and that of the correct signal are always identical, which is independent of the spatial resolution. In order to complete the demonstration of the theory proposed in section 2, instead of using the 30 MHz BFS difference applied on the stressed section, the 10 MHz difference is further performed. Using the three-section dark pulse method, the Brillouin gain spectra with single peak located between the BFS of the uniform fiber (10.732 GHz) and that of stressed section (10.742 GHz) are observed in both spatial resolution scenarios as expected, as the blue solid curves shown in Figs. 12(c) and 12(d). Nevertheless, the correct spectra can be recovered in all cases by using the proposed four-section dark pulse, as the red curves shown in Figs. 12(c) and 12(d). All the curves in Fig. 12 are consistent with the illustrations of mathematic deducing in Figs. 4(a)-4(d).

Fig. 12 The measured Brillouin gain spectra in different scenarios. (a): 1 ns pulse width, 10 cm stressed section with 30 MHz BFS change; (b): 2 ns pulse width, 20 cm stressed section with 30 MHz BFS change; (c): 1 ns pulse width, 10 cm stressed section with 10 MHz BFS change; (d): 2 ns pulse width, 20 cm stressed section with 10 MHz BFS change.

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Since the standard fitting algorithm cannot work well for the double-peak BGS because of its ambiguity, the fitted BFS in case of 30 MHz BFS difference is not shown here. The fitted BFSs of the single-peak BGS (i.e., 10 MHz BFS difference with 10 cm and 20 cm spatial resolutions by using both methods) are shown in Figs. 13(a) and 13(b), respectively. Both methods can detect the stressed section with correct spatial resolution, but the estimated BFS for the three-section pulse method shows 5 MHz error in both spatial resolution scenarios. The results demonstrate that the proposed four-section dark pulse method is capable of providing sub-meter spatial resolution with no impact of the second echo.

Fig. 13 The fitted BFS in different scenarios. (a): 1 ns pulse width, 10 cm stressed section with 10 MHz BFS change; (b): 2 ns pulse width, 20 cm stressed section with 10 MHz BFS change.

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5. Conclusion

In this paper, the negative impact of the second echo phenomenon in BEDS methods has been thoroughly investigated. Making use of the existing analytical model presented in [19], the integrations of the mathematic terms belonging to the correct signal and the unwanted signal imposed by second echo have been carried out over each of their corresponding time regions, respectively. The calculation results have shown that the second echo has exactly the same amount of contribution in comparison with that of the correct signal in time domain, which is independent of the spatial resolution, as long as the spatial resolution is in sub-meter scale. In spectral domain, the impact of the second echo can be simply considered as an unwanted BGS with same peak amplitude but centered at the BFS of the segment neighboring of the correct interrogating region, being superposed on the correct BGS. When the sensing fiber is uniform, the impact is indiscernible. Nevertheless, when a hot spot with the same length as the spatial resolution is deployed, the two equal-amplitude Brillouin gain spectra become center-shifted, manifesting as a synthetic spectrum having one or two peaks, depending on whether the BFS change applied on the hot spot is less than 17 MHz (when the FWHM of the natural BGS is 30 MHz). This behavior could lead to a maximum BFS estimation error of 8.5 MHz, when 17 MHz BFS change is applied. The impact of second echo reduces as the length of hot spot gets longer, since the gain peak of the unwanted BGS contribution gets lower. Therefore, the detrimental impact of the second echo would be serious, when a small strain or temperature change is applied on a short fiber segment. This is an enormous problem for the structural health monitoring, where early detection of the small stress in short section is crucial to prevent potential disaster. A novel four-section pulse based BEDS technique has been proposed in this paper to overcome the second echo effect. A mathematic model has been provided for optimizing the shape, amplitude and width of the third section of the pump pulse, which turns out that the parabolic distribution has the relative better performance. The proposed method keeps most of the features of the BEDS methods (including the disadvantage in terms of the restriction of km-scale sensing range, unfortunately), while annihilating the second echo effect. Therefore, from this point of view, if the one more acquisition time is not concerned, the differential pulse pair technique [18] still has best performance among the methods that have been proposed so far, since it can overcome the problems in terms of both second echo and sensing distance. The feasibility of our method has been validated experimentally with different lengths (10 cm and 20 cm) of a stressed section being applied different BFS change (10 MHz and 30 MHz) for different spatial resolution scenarios (1 ns and 2 ns) in a 10.3 m long fiber compared with the conventional dark pulse technique. The experimental results have not only confirmed the theoretical analysis about the impact of second echo in this paper, but also shown that the proposed technique can recognize a small BFS change (e.g., 10 MHz) on a short segment (e.g., 10 cm) correctly with sub-meter spatial resolution (10 cm), while maintaining the natural width of BGS without using extra measurement time and post-processing algorithm in comparison to the conventional BEDS techniques.

Acknowledgments

This work was supported in part by 863 Program under Contract 2013AA014202, in part by Beijing Municipal Commission of Education, and in part by the BUPT Excellent Ph.D. Students Foundation (CX201431). The authors thank Key Laboratory of Optical Fiber Sensing & Communications (Education Ministry of China) for their valuable research support.

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Evaluating and overcoming the impact of second echo in Brillouin echoes distributed sensing

Abstract

1. Introduction

2. Evaluating the impact of second echo in BEDS methods

3. Proposed four-section pump pulse based BEDS method

3. Experimental results and discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (13)

Equations (19)

Optics Express