Phase-sensitive optical time-domain reflectometry ($\Delta \phi$-OTDR) is a powerful technique for detection, localization, and identification of mechanical perturbations that reach an optical fiber. The range of applications is tremendous, from biomedical devices [1] to the oil and gas industry [2], over sensing dedicated fibers or even smart-city applications [3] exploiting deployed telecom fiber networks. Coherent $\Delta \phi$-OTDR linearly captures phase variations along the fiber induced by mechanical events with a flexible trade-off among spatial resolution, sensitivity, and mechanical bandwidth to adapt to the application field requirements.

However, several forms of fading still stand in the way of optimum sensing performance. While polarization fading can be partially avoided [4–6] or completely mitigated by recently introduced coherent multiple-input, multiple output (MIMO) based on Jones matrix estimates of the backscattered optical field [7,8], Rayleigh fading (also known as speckle noise or coherent fading) remains. Indeed, coherent $\Delta \phi$-OTDR relies on the random constructive and destructive interferences of coherent light inside the fiber sensor [9] such that the sensitivity randomly fluctuates along the fiber sensor and in time. For now, the impact of Rayleigh fading can be limited due to frequency diversity schemes [10,11] at the cost of increased setup complexity.

To mitigate Rayleigh fading noise without increasing measurements or processing time, several groups proposed low-cost approaches either to discriminate the false alarm peaks induced by the fading effect [12] or select a lower resolution set of backscattered points, chosen on a signal-to-noise ratio basis [13]. In these cases, the criterion to decide on the quality of a measurement is based on either on the phase alone or the sole backscattered intensity. In this Letter, we introduce instead, prior to any phase extraction, a criterion based on the Jones matrix estimates of the fiber sensor to early identify the Rayleigh backscatter fading points from the $\Delta \phi$-OTDR trace. Beyond the false alarm identification, the criterion is quantized to derive a soft-bit reliability metric suited for later feeding classifiers and so help distinguish between relevant signal information and Rayleigh fading artifacts.

First, the Letter introduces the reliability criterion derived from the Jones matrix estimates of the backscattered optical field. Second, the criterion relevancy is demonstrated through simulations in a model environment and then through experimental traces with perturbations.

The optical field ${{\textbf E}_{\textbf r}}$ backscattered in a standard SMF fiber can be expressed as the product of transmitted optical sequences ${{\textbf E}_{\textbf t}}$ with $2 \times 2$ Jones matrix ${{\textbf H}_{i,j}}$, which stands for the dual-pass impulse response from the start of the fiber to the $j$th fiber segment at time index $j$ and the injected optical field, as follows: ${{\textbf E}_{\textbf r}} = {{\textbf H}_{{\textbf{i,j}}}}{{\textbf E}_{\textbf t}}$. The channel response is thus fully characterized in terms of intensity, phase, and polarization. A sensing method providing estimations of H was recently introduced [7], based on a joint probing at the transmitter side along two orthogonal polarization axes with mutually orthogonal binary codes modulated in phase; at the receiver side, a coherent mixer captures the backpropagated field over two orthogonal polarization axes as well, hence the name “coherent-MIMO sensing” to highlight the spatial diversity brought by the polarization at both transmitter and receiver sides [13]. It was shown in [8] that this probing technique fully removes both polarization intensity fading and polarization induced phase noise effects.

We recall that the symbol rate ${F_s}$ used to inject the probing code of duration ${T_{\text{code}}} = {N_{\text{code}}.F_s}$ determines the gauge length, or fiber segment length, ${S_r} = {c_f}/(2{F_s})$. ${N_{\text{code}}}$ is the number of symbols that compose the code, and ${c_f}$ stands for the light velocity in the fiber core. The cumulated phase for segment index $i$ is derived from ${{\textbf H}_i}$ as $\frac{1}{2}.\angle \det ({{\textbf H}_i})$. The local phase per segment is then obtained by spatial differentiation of the cumulated phase. By continuously injecting the probing codes, the system provides periodic estimations ${{\textbf H}_{i,j}}$ with a period ${T_{\text{code}}}$ equal to that of the code. Following the Nyquist theorem, the achieved mechanical bandwidth is BW $= 1/(2{T_{\text{code}}})$. In summary, vibrations with a spectral signature below BW-Hz occurring along the fiber can be captured and localized with spatial accuracy ${S_r}$.

Now focusing on the Rayleigh fading effect, let us consider the backpropagated Jones matrix response at segment index $i$, modeled as [8]

The backscattered intensity is derived from the Frobenius norm of ${{\textbf H}_i} = [{\begin{array}{*{20}{c}}{{h_{\textit{xx}}}}&{{h_{\textit{xy}}}}\\{{h_{\textit{yx}}}}&{{h_{\textit{yy}}}}\end{array}}]$ as $|{{\textbf H}_i}|_{\text{frob}}^2 = |{h_{\textit{xx}}}{|^2} + |{h_{\textit{xy}}}{|^2} + |{h_{\textit{yx}}}{|^2} + |{h_{\textit{yy}}}{|^2}$ [13]. Since V is unitary, it is of the form $[{\begin{array}{*{20}{c}}a&b\\{- {b^*}}&{{a^*}}\end{array}}]$, where ${a^*}$ denotes the conjugate of $a$, and $|\det ({{\textbf V}_i})| = 1$. Then $|{{\textbf H}_i}|_{\text{frob}}^2 = |{A_i}{p_i}{|^2}.(|a{|^2} + |b{|^2} + | - {b^*}{|^2} + |{a^*}{|^2}) = 2.|{A_i.p_i}{|^2}$. The intensity can alternatively be derived from the determinant of H by calculating $|\det ({{\textbf H}_i})| = |{A_i}{p_i}{|^2}.(a{a^*} + b{b^*}) = |{A_i}{p_i}{|^2}$. Therefore, the two estimators are equivalent and perfectly estimate the intensity of the backscattered optical field in MIMO sensing:

 

Fig. 1. Comparison of $|{\textbf H}|_{\text{frob}}^2$ and $|\det ({\textbf H})|$ backscattered intensity estimators: (a) analytical study and (b) simulation.

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Given the imperfections of the optical and electronical elements present in a sensing system setup, the estimated matrix $\tilde {\textbf H}$ at the Rx side can be modeled as $\tilde {\textbf H} = {\textbf H} + \epsilon$, with $\epsilon = [{\begin{array}{*{20}{c}}{{\varepsilon _{\textit{xx}}}}&{{\varepsilon _{\textit{xy}}}}\\{{\varepsilon _{\textit{yx}}}}&{{\varepsilon _{\textit{yy}}}}\end{array}}]$ an additive white Gaussian noise (AWGN). Under this assumption, the two intensity estimators may diverge. After analytical development, we get the two following expressions for the Frobenius and determinant estimators, respectively:

Beyond intensity concerns, we recall that phase variations at segment index $i$ are estimated by periodically calculating ${\phi _{\text{MIMO}}} = 0.5.\angle \det ({{\textbf H}_i})$. Therefore, an estimation bias on the matrix intensity also translates into a phase estimation one, giving rise to phase artifacts that degrade the phase sensitivity and that can be wrongly interpreted as mechanical disturbances (false alarms). We reuse the 50 km simulated fiber along with the simulation parameters introduced above. The differential phase is calculated from a subset of fiber segments by selecting, according to both estimators, the highest intensity one among every 10 consecutive segments, leading to an average resolution of 20 m. This technique brings a sensitivity gain at the cost of spatial resolution loss [13]. Figure 2 shows the standard deviation (std) of the phase during the 130 ms simulated period calculated by phase differentiation between the subset of higher intensity segments. In absence of any simulated perturbation, the phase variation is mainly induced by the Rayleigh fading and by the laser coherence loss, this latter effect being responsible for the constant std increase with distance, whereas local variations are induced by randomly distributed Rayleigh fading. The higher intensity segment subset selected from the $|\det |$ criterion shows a much lower std of their differential phase and so a better sensitivity along the 50 km simulated fiber; this is a direct consequence of the lower bias induced by the $|\det |$ intensity estimator relative to the Frobenius norm. Therefore, with MIMO sensing that gives access to the Jones matrix estimation ${{\textbf H}_i}$ of the backscattered optical field at any fiber segment, the determinant module of H is, beyond a fair intensity estimator, also a powerful indicator of the ability to numerically extract a reliable phase estimation.

 

Fig. 2. Impact on phase sensitivity of intensity estimator when selecting a highly reflective segment subset (spatial resolution 100 m). 50 km simulated fiber.

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For sensing methods with polarization diversity at the receiver side only single-input, multiple output (SIMO), the estimated channel comes down to ${\hat{\textbf H}_i} = [{\begin{array}{*{20}{c}}{{h_{\textit{xx}}}}\\{{h_{\textit{yx}}}}\end{array}}] = {A_i}{p_i}[{\begin{array}{*{20}{c}}{{V_x}}\\{{V_y}}\end{array}}]$ [8], where the terms ${V_x}$ and ${V_y}$ depict the change in polarization. We get ${h_{\textit{xx}}} + {h_{\textit{yx}}} = {A_i}{p_i} \times ({V_x} + {V_y}) \ne {A_i}{p_i}$. Therefore, the phase estimation ${\phi _{\text{SIMO}}} = \angle ({h_{\textit{xx}}} + {h_{\textit{yx}}})$ is subject to polarization induced phase noise while the intensity one, defined as ${{\tilde {\rm I}}_{\text{SIMO}}} = |{h_{\textit{xx}}}{|^2} + |{h_{\textit{yx}}}{|^2}$, encloses the Rayleigh fading. Note that this is the only way here to estimate the backscattered intensity level. Nevertheless, ${{\tilde {\rm I}}_{\text{SIMO}}}$, though less accurate than the MIMO estimator due to a lack of channel estimation under a Jones matrix form and so subject to the polarization fading effect, can also be exploited as an indicator of the ability to extract a reliable phase estimate for SIMO implementations.

The previous section, based on analytical development and simulations, has shown the relevancy of the $|\det ({\textbf H})|$ metric in MIMO sensing to enhance the backscattered intensity estimation at low-reflective fiber segments compared to the more usual Frobenius norm. The finer discrimination between low intensity segments was shown to select a more relevant segment subset, thus also improving the estimated phase along the tested fiber due to higher sensitivity. This section first aims to experimentally confirm the benefit on the phase sensitivity.

Figure 3 displays the overall experimental setup. The fiber under test (FUT) is probed over two orthogonal polarization axes due to a Mach–Zehnder modulator that modulates a 1536.1 nm ultra-narrow linewidth (75 Hz at low frequencies [13]) laser source with the same interrogation setup (codes and baud rate) as in the simulation part. In a first experiment, we probe a 50 km length standard telecom fiber spool placed in a mechanically insulated box to make it less sensitive to environmental noise. The Rayleigh backscattered signal is captured by a coherent mixer, electrically converted, and then digitized. After a correlation process, a set of Jones matrices estimating the backscattered optical field in both time and distance dimensions is available. The differential phase is finally derived from a subset of highly reflective segments according to both Frobenius and $|\det |$ criteria in the same way, so to achieve the same spatial resolution as in the above simulated case.

 

Fig. 3. Experimental MIMO sensing setup.

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Figure 4 displays, as with Fig. 2, the std in time of the phase along the 50 km FUT. The phase std is slightly higher than in the simulation case mainly due to a lack of insulation at low mechanical frequencies. However, we observe the same trend here: the segment selection made with the |det| intensity criterion brings on a final basis a much lower rate of artifacts than with the Frobenius one and so an enhanced sensitivity.

We have until now considered the std of the phase per fiber segment in absence of external perturbation. This is based on the statement that, in such a static case, both intensity and phase remain quite stable at any segment during the observation time. Let us now focus on a dynamic case with the FUT mechanically excited at a segment location. A 1 km SMF is locally excited by a 15 Hz acoustic sine wave at 640 m from the fiber start. The sound pressure level reaching the fiber could not be accurately measured at such a low frequency. However, the captured phase variation was sufficiently low to assume that polarization parameters are left unchanged and so that it affects the fiber length only locally.

Figure 5 shows the variations in time of the differential phase measured at the excited segment position. The sinewave excitation is quite well captured by the phase though some distortion can be observed, with local artifacts. Superimposed is the intensity estimated by the $|\det |$ metric, normalized over all the ${{\textbf H}_{i,j}}$ of the measurement to set the highest estimated intensity $\mathop {\max}\nolimits_{i,j} (|\det ({{\textbf H}_{i,j}}|))$ to one. First, the $|\det |$ metric is modulated by the captured mechanical excitation. This was expected since the applied excitation, when expanding the length of the excited fiber segment, slightly modifies the initial position of the elementary scatterers and so modulates the intensity term $|{A_i}{p_i}{|^2}$. Second, there is a correlation between the position of the artifacts observed in the phase response and the local minima of the $|\det |$ metric. This correlation is not perfect since the $|\det |$ intensity estimator is slightly biased at low intensity levels, and so false positives or negatives cannot be excluded. Nevertheless, it highlights that such artifacts are erroneous phase estimates that occur when the Jones matrix intensity drops.

 

Fig. 4. Impact on phase sensitivity of intensity estimator when selecting a highly reflective segment subset (spatial resolution 20 m). Experimental measurement, no excitation.

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Fig. 5. Phase variations with fading artifacts observed at a fiber segment perturbated by a 15 Hz sinewave (black, bottom) and associated normalized $|\det |$ estimator (gray, top).

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Therefore, the $|\det |$ metric can be exploited in MIMO sensing not only as a fair intensity estimator but also as a criterion that informs about the ability to extract from any Jones matrix H reliable phase information. The normalized intensity estimator $|\det ({{\textbf H}_{i,j}})|/\mathop {\max}\nolimits_{i,j} (|\det ({{\textbf H}_{i,j}})|)$ appears in Fig. 6, superimposed to the phase plot under the form of a color code to inform about the confidence level of each extracted phase term. This soft-decision metric can be conveniently used in practical situations to distinguish between true mechanical information and artifacts induced by the phase extraction process. Also, it may advantageously feed a further artificial intelligence based automatic recognition system to enhance the classification process of the captured waveforms.

 

Fig. 6. Phase variations in time at segment location in Fig. 5. Superimposed is color coded $|\det |$ criterion to highlight the artifact detection.

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We have highlighted the relevance of a correct estimation of the backscattered intensity in ϕ-OTDR applications. MIMO sensing gives access to the Jones matrix of the backscattered optical field, and we have demonstrated that its determinant module introduces a smaller bias than the standard Frobenius norm to estimate the intensity. In a first step, it has allowed to better discriminate between low-reflective segments along the sensed fiber, thus enhancing the phase sensitivity. In a second one, we have shown that this determinant module can be used as a soft-decision reliability metric to detect phase artifacts induced by local intensity drops that potentially superimpose to any detected mechanical signal. This metric is of major interest for artificial intelligence based post-processing to help improve the classification of the captured mechanical signals.