Abstract
Distributed optical fibre sensors deliver a map of a physical quantity along an optical fibre, providing a unique solution for health monitoring of targeted structures. Considerable developments over recent years have pushed conventional distributed sensors towards their ultimate performance, while any significant improvement demands a substantial hardware overhead. Here, a technique is proposed, encoding the interrogating light signal by a singlesequence aperiodic code and spatially resolving the fibre information through a fast postprocessing. The code sequence is once forever computed by a specifically developed genetic algorithm, enabling a performance enhancement using an unmodified conventional configuration for the sensor. The proposed approach is experimentally demonstrated in Brillouin and Raman based sensors, both outperforming the stateoftheart. This methodological breakthrough can be readily implemented in existing instruments by only modifying the software, offering a simple and costeffective upgrade towards higher performance for distributed fibre sensing.
Introduction
Distributed optical fibre sensors (DOFS)^{1} offer the capability to continuously inform on the spatial distribution of environmental quantities (such as temperature/strain^{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}, acoustic impedance^{17,18}, refractive index^{19}, etc.) along with an optical fibre. Most DOFS exploit natural scattering processes present in optical fibres^{20}, such as Rayleigh^{2,3,4,10}, spontaneous Raman^{5,6,11} and spontaneous/stimulated Brillouin^{9,12,13,14,15,16,17,18,19} scatterings. These backscattered optical signals can be interrogated in time^{2,3,4,5,6,7,8,16,17,18}, frequency^{9,10,11}, correlation^{12,13,19} or mixed^{14,15} domains, each with their proper advantages targeting specific applications^{1}.
Timedomain approaches based on optical timedomain reflectometry (OTDR)^{21} turn out to be intrinsically best suited for longdistance sensing (e.g. tens of km), since the sensing range is limited by the laser coherence length in frequencydomain approaches^{1,10}, and by background noises originating from uninterrogated fibre positions in correlationdomain approaches^{1,12}. Timedomain approaches enable to spatially resolve the local environmental information by analysing the backreflected response from an intense optical pulse. The pulse energy is positively related to the measurement signaltonoise ratio (SNR), which in turn ultimately scales and tradesoff all critical sensing specifications (i.e. spatial resolution (SR), measurand accuracy, sensing range and measurement time)^{1,22,23,24}. For any conventional timedomain DOFS, the maximum attainable SNR turns out to be limited by fundamental physical constraints: whilst the maximum pulse peak power is limited by the onset of nonlinear effects^{25,26,27,28,29}, the minimum noise power is determined by the signal bandwidth thus being irreducible whatever the denoising technique used^{30,31}.
To further improve the optimised performance of conventional timedomain DOFS, the energy of the sensing response must be enhanced without impairing the SR nor activating nonlinear effects. This can be realised by using advanced techniques, such as distributed or lumped optical amplification^{32,33,34,35}, smart signal modulation^{4,16,36,37,38,39}, optical pulse coding^{40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61} and any combination of them^{62,63}. Among these approaches, optical pulse coding is conceptually the most costeffective, which operates by launching optical pulse trains into the sensing fibre and demodulating (decoding) the backscattered multipulse response through postprocessing, delivering an SNRimproved equivalent singlepulse response. Extensive studies have been carried out to maximise the coding operation efficiency^{40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64} (i.e. maximising the SNR improvement, socalled coding gain, whilst minimising the hardware overhead and extra measurement time); however, all currently existing code types present fundamental and/or practical limitations. For instance, although codes formed by a single sequence^{40,56,57,58,59,60,61,62,64} are ideal for minimising measurement time and data storage, they either exhibit an imperfect peaktosidelobes ratio after the decoding^{40,56,64} or are periodic/cyclic codes^{57,58,59,60,61,62,64} that suffer from impairments induced by signaldependent noise^{1} (resulting in a compromised coding gain that has not yet been quantified in literature). As more robust alternatives, distortionfree aperiodic codes constituted by several sequences, such as Golay^{41,42,43,44,45,46,47}, Simplex codes^{48,49,50,51,52,63} and their derivatives^{54,55}, have been mostly used for DOFS. Both codes can provide SNR enhancement with the same number of total acquisitions as in singlepulse DOFS; however, the additional operation time (e.g. the decoding time and the codeword switching time) and the larger occupied memory (for storing distinct coded traces) compromise the coding operation efficiency. More importantly, to reduce the decoding distortion, additional devices are used to alleviate the decaying envelope of the pulse sequences imposed by the gain saturation of erbiumdoped fibre amplifiers (EDFAs)^{46,50,53}. Nevertheless, the unevenness of the amplified pulse train cannot be fully suppressed^{53}, and remaining distortions are still detrimental for certain types of codes (e.g. Golay codes^{53}). All abovementioned issues make the conceptual advantages of the coding technique to be hardly realised in practice to match the vast diversity of realworld applications.
Here, we propose an approach based on the concept of deconvolution, offering the possibility of a singlesequence aperiodic code and overcoming all abovementioned drawbacks inherent to conventional codes. This enables, for the first time to the best of our knowledge, substantial performance improvement over optimised singlepulse DOFS without any hardware extension. The aperiodic code sequence is once forever optimised by a dedicatedly developed distributed genetic algorithm (DGA), here designated as geneticoptimised code (GOcode), offering a coding gain comparable to that of conventional codes such as Golay and Simplex. More importantly, the flexibility in the code length and the adaptability to the nonuniform amplitude over the pulse sequence (namely the pulse sequence envelop) secure a maximum coding efficiency for any given experimental condition, whilst the minimised operation time and the fast decoding process enables online realtime measurements, all being crucially advantageous over conventional codes. The proposed technique can be readily implemented in any classical timedomain DOFS by only modifying the software in relation to the pulse generation and postprocessing (decoding), showing an unmatched practical significance. All these advantages are here experimentally validated using the most standard implementations of distributed sensors based on Brillouin optical timedomain analysis (BOTDA) and Raman OTDR (ROTDR). It must be emphasised that all demonstrations for both GOcode and singlepulse schemes are performed under fully optimised conditions, i.e. using maximised signal powers^{25,26,28} and minimised noise bandwidth (by matching the noise bandwidth to the signal bandwidth through digital filtering^{31}), thus securing fair comparisons.
Results
Description of the proposed coding technique
The proposed technique can retrieve the singlepulse response with an improved SNR by only modifying the software of a standard singlepulse distributed sensor in relation to pulse generation (upper row in Fig. 1) and postprocessing (bottom row in Fig. 1). In this case, an N_{c}point discretetime signal c(n), representing the digitally coded pulse sequence, is uploaded into a programmable pulse generator (e.g. fieldprogrammable gate array, FPGA), which in turn converts c(n) into an electrical signal driving the optical setup. The integer n is linearly related to the time t through the sampling rate f_{s}, so as t = n/f_{s}. The digital coding sequence c(n) is once forever generated from a N_{u}point binary unipolar sequence u(n) following the three steps described in Fig. 1. Here, u(n) contains 0’s and 1’s elements and must be dedicatedly designed, as will be elaborated in the next subsection.
Although the amplitude of c(n) is uniform in the digital domain, the coded optical pulse sequence launched into the fibre (blue dots in the middle row of Fig. 1), designated as c_{f}(n), may exhibit a nonuniform amplitude envelop (grey dashed curve in the middle row of Fig. 1) due to the gain saturation of the EDFA used to boost the optical sequence power. Note that all sequences defined hereafter with a subscript ‘f’ are associated with such a nonuniform amplitude envelope. The acquired coded fibre response \(r_c^m(n)\) can be commonly expressed by the linear convolution^{65} between the optical code sequence c_{f}(n) launched into the fibre and the fibre impulse response h(n), merged with a zeromean additive noise e_{c}(n)
Defining the fibre length L_{h}, the number of samples in the fibre impulse response h(n) is N_{h} = 2L_{h}f_{s}/v_{g}, which further yields the number of samples in the coded fibre response \(r_c^m(n)\) and the noise component e_{c}(n) to be N_{r} = N_{c} + N_{h} − 1. In Eq. (1) the coded optical sequence c_{f}(n) can be considered as the linear convolution between an N_{d}point sequence d_{f}(n) and the single pulse signal p(n), where d_{f}(n) = f(n)d(n), f(n) being the envelope function of c_{f}(n) determined by the EDFA gain saturation, as shown by the grey dashed curve in the middle row of Fig. 1. This way, Eq. (1) and its discrete frequencydomain representation can be expressed as
where F denotes the N_{r}point discrete Fourier transform (DFT) operation^{65}.
Here, we designate D_{F}(k) as the decoding function, which is known since its timedomain representation, d_{f}(n), can be precisely and easily obtained by retrieving the energy of each optical pulse in the optical code sequence c_{f}(n) measured at the fibre input (see middle row of Fig. 1). Such calibration can be made for a long measurement session, provided that the experimental stability is sufficient and there is no uncontrolled signal drift of significant importance in the instrumentation. Based on Eq. (2) and inverse DFT^{65}, the targeted singlepulse response r_{s}(n) (=p(n)⊗h(n)) can be retrieved by performing the following decoding process:
where \(r_s^d(n)\) is the decoded singlepulse response that contains the targeted singlepulse response r_{s}(n) and a noise term affected by the decoding function D_{F}(k). Equation (3) highlights that the nondistorted singlepulse fibre response r_{s}(n) can be recovered regardless of the optical coding sequence distortion introduced by the EDFA gain saturation, unlike other conventional codes (e.g. Simplex and Golay) that strictly require a uniform optical code sequence, i.e. f(n) ≡ 1. In addition, the decoding process can be made very fast since the DFT can be computed via fast Fourier transform^{65}.
In Eq. (3), the measure of whether the D_{F}(k) attenuates or magnifies the original input noise is given by a noise scaling factor Q (for details see Supplementary Eqs. (8)–(12)), which represents the ratio of the noise variance after (\(\sigma _{de}^2\), see Supplementary Eq. (8)) and before decoding (\(\sigma _e^2\), see Supplementary Eq. (4))
where U_{F}(k) is the N_{r}point DFT of the sequence u_{f}(n) (with an amplitude envelope given by the EDFA gain response), the latter being obtained by an N_{x}point downsampling of d_{f}(n), as shown in the middle row of Fig. 1. This way the coding gain (usually defined as the reduction in the noise standard deviation, i.e. σ_{e}/σ_{de}, like all other codes for DOFS) resulting from our proposed code is \(G_c = \sqrt {1{\mathrm{/}}Q} \). It appears evident that to maximise the coding gain, it is required to dedicatedly design and optimise u_{f}(n).
Design and optimisation of the code
The design and optimisation of u_{f}(n) should firstly maximise the optical energy increment provided by the coded optical pulse sequence c_{f}(n) with respect to the single optical pulse p(n). Such energy increment is represented by an energy enhancement factor \(F_E = \mathop {\sum}\nolimits_{n = 1}^{N_u} {u_f\left( n \right)} \), which is a generic parameter valid for any type of coding. Note that F_{E} equals to the number of coded pulses only when the code assumes a uniform amplitude envelope (the case of other conventional codes). In general, the upper bound of F_{E} for any code is ultimately limited by the onset of undesired physical phenomena that get more severe for a larger F_{E}, such as highorder pump depletion^{53} and/or additional Brillouingain dependent noises (see Supplementary Note 3) in BOTDA, and amplified shotnoise in ROTDR (see Supplementary Note 4). However, for other conventional codes, F_{E} (the number of coded pulses) is additionally constrained by their mathematical rules^{42,48}, so that the selected F_{E} is restricted to specific values that could be much smaller than the ideal upper bound. This limitation makes them less flexible and efficient than the proposed code.
After determining the optimum F_{E} (the above mentioned upper bound value), the code length N_{u} should be adapted accordingly (i.e. the larger the F_{E}, the longer the N_{u}). This essentially means optimising the ratio m = N_{u}/F_{E} (m ≥ 1), which is related to the theoretical maximum coding gain G_{c} (derived based on Eq. (4), see Supplementary Eqs. (12)–(19)):
To evaluate G_{c}, Eq. (5) is compared with standard reference coding gain \(G_r = \sqrt {F_E{\mathrm{/}}2} \) offered by other conventional unipolar codes commonly used in DOFS, thus further yielding
It must be mentioned that not all standard reference coding gain G_{r} can be practically realised since not every value of F_{E} is allowed by conventional codes as mentioned before, hence here G_{r} only represents a numerical reference to evaluate the performance of the proposed technique. Equation (6) suggests that the larger the value of m, the larger the upper limit of G_{c}/G_{r}, as illustrated by Fig. 2a in logarithmic scale. To make G_{c}/G_{r} closest possible to this upper limit, the detailed code distribution in u_{f}(n) must be dedicatedly designed. This is equivalent to design 1’s and 0’s distribution in u(n), since u_{f}(n) equals to u(n) being modulated by a known decaying envelope (see Fig. 1).
It turns out that the ideal design should make the offpeak spectral region of U_{F}(k)^{2} (the power spectrum of the designed code) as flat as possible for given F_{E} and m, as elaborated in Supplementary Note 1, which is extremely complex and cannot be analytically solved to the best of our knowledge. As an alternative, here a DGA^{66} is dedicatedly developed to efficiently search for optimal solutions, so that the code proposed here is designated as GOcode. The general concept and detailed implementation of the DGA are introduced in Methods, and its performance is elaborated in Supplementary Note 2 along with some intermediate results. The optimum value of m for a given energy enhancement factor F_{E} is empirically found to be between 3 and 4, as illustrated in Fig. 2b. The reason comes from the fact that the high computational complexity in the case of long code lengths (large values of F_{E} and m) leads to a reduction of the genetic optimisation efficiency. Following this criterion, the optimum values of G_{c} achieved by the DGA searching as a function of F_{E} are shown in Fig. 2c for m = 3 (note that similar curves have also been obtained for m = 4, showing only minor differences in terms of gain). It can be found that when F_{E} = 20, the coding gain G_{c} is 0.3 dB larger than the standard reference coding gain G_{r}; however, the larger the energy enhancement factor F_{E}, the more demanding the computational capability, thus leading to a larger compromise of G_{c} with respect to G_{r}. Nevertheless, the maximum difference is always <0.5 dB, which can be considered negligible in most applications.
GOcoded BOTDA
The performance of the proposed GOcode for BOTDA is investigated and compared with an optimised standard singlepulse BOTDA using the same experimental configuration (see “Methods”). In the setup, the singlepulse and GOcode schemes can be readily switched by alternating the pulse and coding sequence driving the FPGA. For both schemes, the SR is set to 2 m, corresponding to a pulse duration of 20 ns and an optimised singlepulse peak Brillouin gain of 2.5% at the fibre nearend. Other detailed experimental parameters are listed in Table 1.
For the GOcode scheme, the optimum energy enhancement factor F_{E} is determined to be ~40, to ensure that the coding gain at the fibre farend is not degraded by additional Brillouingaindependent noises (see details in Supplementary Note 3). Then, following the procedure described in Algorithm 1, and using parameters listed in Table 2, a 135bit GOcode sequence is found by the proposed DGA. According to Eq. (4), such GOcode sequence is expected to offer a coding gain of G_{c} = 6.2 dB, being only 0.3 dB lower than the standard reference coding gain (G_{r} = 6.5 dB, \(\sqrt {40{\mathrm{/}}2} = 4.47\) times in linear scale). The GOcoded optical pulse sequence measured at the fibre input with a duty cycle of 20% is represented by the blue curve in Fig. 3a. Note that the actual peak power of the first pulse has been adjusted equal to that of optimised single pulse for a fair comparison, and is here normalised to 1 in the figure for the sake of clarity. The corresponding d_{f}(n) used for decoding is precisely retrieved based on the actual envelop of this sequence, as the red curve shown in Fig. 3a. Using such d_{f}(n) and based on Eq. (3), decoded singlepulse responses for each scanning frequency are obtained, with a total decoding time of 1.8 s that has no critical impact on the total measurement time (~2.8 min).
The decoded singlepulse response at the Brillouin resonance is shown by the red curve in Fig. 3b, together with BOTDA traces obtained from the singlepulse scheme using 1024 (blue) and 17795 (black) averages, respectively. The latter case is set as a reference, whose SNR improvement with respect to the 1024averaged singlepulse measurement is equivalent to the one that could be ideally obtained by the designed GOcode scheme (\(10lg\sqrt {17,795{\mathrm{/}}1024} = 6.2\,{\rm{dB}}\)). The figure demonstrates that the decoded response remains undistorted compared to the reference trace, validating the adaptability of the GOcode scheme to an uneven pulse power distribution along the sequence. Due to the additional Brillouingaindependent optical noises in the GOcoded system (see Supplementary Note 3), a clear difference in the noise behaviour between the decoded response and the reference trace (visualised through the thickness of all traces) can be observed, as quantified by the SNR profiles shown in Fig. 3c. The impact of these noises reduces with distance due to the fibre attenuation, and eventually becomes negligible (i.e. showing a tiny impact on the expected coding gain) near the fibre farend thanks to the proper optimisation of F_{E}. Although this behaviour has not yet been clearly documented, it is evident that such enhanced Brillouingaindependent optical noises are inherent to any codedBOTDA, being a nonexclusive effect in the GOcode proposed here, as experimentally verified for Simplexcoded BOTDA in Supplementary Fig. 9.
From the measured Brillouin loss spectrum, the Brillouin frequency shift (BFS) profile along the sensing fibre is estimated through a crosscorrelation method^{67}. The obtained BFS profile around a 5m long hotspot placed at the fibre farend is shown by the black curve in Fig. 3d, which matches well with a reference curve (blue) obtained from the singlepulse scheme with 2 m SR and 17,795 averages, and outperforms the singlepulse measurements with 1024 averages (red). To eventually verify the performance of the proposed GOcode, the BFS uncertainty profiles are calculated over five consecutive measurement for both singlepulse (blue) and GOcode (red) schemes, as shown in Fig. 3e. The frequency uncertainty at the fibre farend obtained by the GOcode scheme (0.63 MHz) compared to the singlepulse scheme (2.65 MHz) is 4.1fold enhanced, in good agreement with the theoretically evaluated coding gain (6.2 dB).
A similar comparison is then carried out at 1 m SR, which allows for a larger optimal energy enhancement factor F_{E} = 200 due to the smaller optimised singlepulse Brillouin gain (0.5%) at the fibre nearend. In this case a 723bit GOcode sequence is found by DGA, offering a theoretically evaluated coding gain of 9.3 dB, being 0.7 dB lower than the reference coding gain. Whilst behaviours similar to the case of 2 m SR are observed (see Supplementary Fig. 13a–c), an 8.59fold BFS uncertainty reduction (from 18.9 MHz for singlepulse scheme to 2.2 MHz for GOcode scheme) is demonstrated, as shown in Fig. 3f, matching perfectly the theoretically calculated coding gain of 9.3 dB.
GOcoded ROTDR
In this section, the performance of the proposed GOcode for ROTDR is investigated and compared with an optimised singlepulse case, both based on the same experimental layout (see “Methods”).
Comparative measurements with 2 m SRs are carried out over a 39 km sensing range. A standard singlemode fibre (SMF) is used to eliminate any multimodal dispersion, thus avoiding SR impairments near the fibre farend^{1}. For both singlepulse and GOcode schemes, each temporal trace is acquired over 0.408 ms and averaged 2 million times, leading to a practical measurement time of 13.6 min. In the GOcode scheme, RZ format with a duty cycle of 22.2% is used to avoid the interaction between pulses in the code sequence and their forward Raman scattering components^{57}. The energy enhancement factor F_{E} for the GOcode ROTDR scheme is limited by the contamination of the amplified shotnoise (as elaborated in the Supplementary Note 4), and its optimal value is calculated to be F_{E} = 44. Taking into account the decaying envelop of the pulse sequence imposed by the EDFA gain saturation, the DGA delivers a 177bit GOcode with a theoretical coding gain G_{c} = 6.37 dB, being only 0.34 dB lower than the standard reference coding gain G_{r} = 6.71 dB. The normalised GOcoded pulse sequence measured at the EDFA output is shown by the blue curve in Fig. 4a, along with the reconstructed d_{f}(n) (red curve) that is required for decoding. The decoding time is found to be 10 ms, showing a negligible impact on the total measurement time as well.
The resolved temperature profiles obtained from the singlepulse and GOcode schemes are shown by the blue and red curves in Fig. 4b, respectively, indicating the clear performance enhancement provided by the proposed GOcode. Such a performance enhancement is then quantified by computing the corresponding SNR profiles, as shown in Fig. 4c. A coding gain of 6.33 dB at the fibre farend is demonstrated, being in good agreement with theory. As anticipated, the SNR improvement at the fibre nearend is lower than the theoretical value due to the signalpowerdependent amplified shotnoise in the avalanche photodetector (APD), as elaborated in Supplementary Note 4. To verify the SR achieved by the GOcode scheme with ROTDR, the temperature of a 5mlong hotspot introduced at the fibre farend has been increased up to 70 °C, while the rest of the fibre is kept at the room temperature (25 °C). The temperature profiles obtained using the GOcode scheme is shown by the red curve in Fig. 4d, matching well with the trend of a reference temperature profile (blue curve) obtained by the singlepulse scheme with 2 m SR. Finally, Fig. 4e demonstrates a 4.27fold temperature uncertainty enhancement at the fibre farend (from 8.3 °C in the singlepulse case down to 1.9 °C with GOcode) thanks to the GOcode coding gain.
Measurements with 1 m SR are then performed with the same sensing fibre and measurement time. In this case, the optimised energy enhancement factor F_{E} is determined to be 150, leading to a 450bit code that takes into account the decaying trend of the sequence envelops. The GOcode delivered by the DGA exhibits a coding gain of G_{c} = 8.67 dB. Relevant results associated with 1 m SR are illustrated in Supplementary Fig. 14a–c. Benefiting from the coding gain, the GOcode scheme offers 3.9 °C temperature uncertainty at the fibre end, representing a 7.24fold improvement with respect to the uncertainty of 28.2 °C obtained by the singlepulse scheme, as shown in Fig. 4f.
In addition to the advantage of simple implementation, the capability of fast temperature sensing enabled by the proposed GOcode is demonstrated with 2 m SR and a 10.2km SMF, using the same 177bit code as illustrated in Fig. 4a. Temporal traces are acquired over 0.12 ms and averaged 9000 times, leading to a practical measurement time of ~1 s and decoding time of 1.6 ms. It must be mentioned that this short measurement and decoding time is indeed an exclusive feature of the proposed code, and cannot be reached by any other conventional codes. Due to the APDinduced amplified shot noise, the SNR improvement at the fibre farend is found to be 5.5 dB, as quantitatively shown by the SNR profiles at 10.2 km in Fig. 4c. This corresponds to a 3.5fold temperature uncertainty reduction, as can be observed in Fig. 4e (at 10.2 km, the GOcode reduces the temperature uncertainty from 8.3 °C in the singlepulse case down to 2.4 °C).
Based on these characterisations, a fast online realtime measurement that monitors the temperature of water under heating (from 25 to 100 °C) is performed. Figure 5a shows the twodimensional map of the retrieved temperature as a function of the fibre position and the acquisition time, in which the temperature evolution of the hotspot (at 10.176 km) under heating can be identified. Figure 5b shows the measured evolution of the hotspot temperature (red) as a function of time, in good agreement with that measured by the singlepulse scheme (blue) and reference curve (black, measured by a highprecision thermometer). Figure 5c shows the temperature difference between measurements and reference, verifying the 3.5fold performance enhancement. To explicitly show the fast decoding capabilities of the proposed GOcode scheme, a video is included in the Supplementary Material, displaying a realtime measurement of fast heating and cooling processes over two fibre sections (2 and 5 m, separated by a 5 m unperturbed section) at the fibre farend. In the video, figures are obtained by online realtime decoding (synchronised with the measurements).
Discussion
Whilst providing an SNR improvement comparable to sophisticated stateoftheart techniques, the intrinsic mechanism of the proposed GOcode leads to overall advantages listed in Table 3:

1.
the launch of only one sequence into the sensing fibre, minimising operation time and data storage. Note that Cyclic codes share the same advantage, while Simplex and Golay codes demand additional codeword switching time and larger memory usage, as indicated in the two top rows of Table 3;

2.
a negligible postprocessing (decoding) time (e.g. 1.6 ms decoding time with respect to the measurement time of 1 s for ROTDR), enabling realtime online fast sensing. The decoding time for each coding technique is summarised in the third row of Table 3, in which it can be found that the Cyclic coding has the fastest decoding process. For instance, in the case of a 100 kmlong sensing range using a sampling rate of 250 MS/s and a 255bit code (i.e. N_{h} = 250,000, N_{u} = 255), the evaluated decoding time of Cyclic is three times shorter than that of the proposed GOcode;

3.
an aperiodic structure avoiding detrimental effects existing in singlesequence periodic (cyclic) codes, as indicated in the fourth and fifth rows of Table 3:

a.
the impaired SNR all along with the sensing fibre due to the additional signaldependent noise originating from the first kilometres of the fibre (theoretically justified in Supplementary Notes 3 and 4). This leads to a largely compromised coding gain at all fibre positions, as experimentally demonstrated using BOTDA in Supplementary Note 3;

b.
the absence of the continuous background observed in the measured cycliccoded response, hindering any signal normalisation^{53} required before decoding (the signal normalisation is an essential process before decoding to extract the linear coded response from the measured data^{53});

4.
an adaptability to a nonuniform power over the pulse sequence (e.g. resulting from EDFA gain saturation), as indicated in the sixth row of Table 3, thus avoiding any modification of the instrumental layout compared to an optimised singlepulse implementation. This represents the ultimate asset of the technique, in full contrast with Simplex and Golay codes that require additional devices (higher cost) or predistortion operation (more complexity) to alleviate the unevenness of the coded sequence imposed by the EDFA gain saturation. Even with an improved uniformity (the amplitude envelope of the sequence is never perfectly uniform), decoding distortions remain still present when using Golay codes^{53}, resulting in a degraded performance;

5.
enabling the use of any number of pulses in the code sequence, thus securing the highest coding efficiency in any given condition.
Benefiting from all these advantages, the proposed GOcoding technique enables, for the first time to the best of our knowledge, a pure overall performance improvement over optimised conventional DOFS without any hardware modification. The method only requires a software modification for sequence generation and postprocessing, providing a fully compatible and costeffective solution to improve the performance of existing instruments.
The performance of the proposed GOcode has been experimentally demonstrated by comparing with fully optimised singlepulse BOTDA and ROTDR (i.e. using a maximised signal power^{25,26,28}, and minimised noise bandwidth by applying digital Gaussian filtering^{31}). In the case of Raman sensing, further improvement in SNR could be achieved by replacing the sensing fibre by a fewmode fibre^{1,68,69}, which enhances the allowable peak power of the pump light, with a moderate penalty in the SR at mediumlong distances.
It is worth pointing out that the SNR enhancement provided by the GOcode and its features of singlesequence and fast decoding can lead to a huge reduction of the measurement time while maintaining other specifications. For instance, the 8.7fold SNR enhancement demonstrated in the implemented GOcoded BOTDA sensor with 1 m SR can be used to reduce the number of traces averaging, leading to a measurement time reduction of factor 75 (=8.7^{2}), while keeping the same spatial and measurand resolutions as the singlepulse BOTDA implementation for the same sensing distance. This means that a singlepulse measurement of 5.7 min can be performed by a GOcoded system in only ~5 s, thus leading to a significant enhancement in the response time of the sensor with no penalty on other specifications.
It must be mentioned that the code sequences found by the DGA may not be optimal, which is a common feature shared in most searching algorithms^{66,70}. In addition, due to the high computational complexity when searching long code sequences, the finite computational capability limits the searching efficiency. This means that the theoretical maximum coding gain indicated in Fig. 2a, which may even exceed the gain provided by conventional codes (so far only being observed in the case of small F_{E} as shown in Fig. 2c), is still possible to be reached if the searching is carried out with more powerful tools and approaches.
Note also that, although the code studied here has been designed in unipolar format (involving 0’s and 1’s elements), the DGA can be readily adapted to search for codes in bipolar format (involving −1’s and 1’s elements) that can be applied to phaseencodable DOFS, such as phasesensitive optical timedomain reflectometers (ϕOTDR)^{1,2}. Furthermore, the presented approach can benefit not only DOFS but also diverse fields in technical science where the retrieval of the singlepulse response with high SNR is required, such as incoherent OTDRs for fault detection, laser range finders, lidar systems, among others.
Methods
BOTDA setup
The experimental configuration used for GOcoded BOTDA is shown in Fig. 6, where the hardware layout is strictly the same as that for a standard singlepulse BOTDA. In detail, a continuouswave (CW) light from a distributed feedback laser operating at 1551 nm is split into probe and pump branches by a 50/50 coupler. In the probe branch (upper arm in the figure), the light travels through an electrooptic modulator driven by a microwave signal, producing a carriersuppressed double sideband modulated probe wave. The frequency of the driving radiofrequency signal is scanned to reconstruct the BGS along the sensing fibre. This dualsideband probe is launched into a 200 kmloop configuration^{63} with an optical power of 3 dBm/sideband. This loop configuration consists of two fibres: the 100 kmlong leading fibre delivers a probe wave with −18 dBm/sideband to the farend of the other 100 kmlong sensing fibre that acts as a distributed sensing element, thus realising a real remoteness of 100 km for the sensing range^{63}. An isolator is placed between the leading and sensing fibres to block the pump wave, thus restricting the Brillouin interaction only to the sensing fibre. In the pump branch (lower arm in the figure), the light is intensitymodulated to a single optical pulse or a coded optical pulse train, using a high extinction ratio (>50 dB) semiconductor optical amplifier (SOA) driven by a commercial FPGA. Preceded by an EDFA and a tuneable attenuator to optimise the pump wave power to ~23 dBm (determined by the onset of MI^{28}), a polarisation scrambler is inserted to mitigate polarisation fading and polarisation pulling effects^{53}.
In the receiver part, an EDFA operating in lineargain regime is used between two tuneable narrowband fibre Bragg gratings (FBGs). The FBG1 selects the highfrequency probe sideband for the detection (i.e. making the system to operate in Brillouin loss configuration), thus preventing the EDFA from being saturated by unwanted frequency components, such as the Rayleigh backscattering and the second probe sideband. The FBG2, having the same filtering window as that of the FBG1, is used to filter out the amplified spontaneous emission (ASE) noise introduced by the EDFA, and thus to mitigate the ASE–ASE beating noise at the photodetection. The selected optical signal is finally detected by a 125 MHz photodetector and acquired by a realtime data acquisition system with a sampling rate of 250 MS/s.
ROTDR setup
Similar to the BOTDA case, the experimental configuration used for GOcoded ROTDR is strictly the same as that for standard singlepulse ROTDR, as shown in Fig. 7. A CW light from a broadlinewidth (1.2 nm) light source operating at 1551 nm is injected into a high extinction ratio (>50 dB) SOA, driven by a commercial FPGA, to modulate the light intensity with a single optical pulse or a GOcoded pulse sequence, alternatively. The lightwave is then amplified by a highgain EDFA to a peak power of 1 W and launched into the sensing fibre through a wavelength division multiplexer (WDM). The backscattered spontaneous antiStokes (AS) Raman signal is extracted by the same WDM and sent to an APD with a bandwidth of 125 MHz followed by a transimpedance amplifier. The photoelectrical signal is acquired by a realtime data acquisition card with a sampling rate of 250 MS/s.
Note that such a layout corresponds to a simplified ROTDR setup that makes use of a single APD and acquisition channel to measure only the AS spontaneous backward Raman scattering. This simplified setup allows us carrying out a proofofconcept experiment in labcontrolled conditions. Wavelengthdependent losses are made negligible, so that the Raman Stokes or Rayleigh component is not required for realtime compensation. Instead, a simple precalibration procedure that measures the precise AS intensity at room temperature is performed to compensate for local and distributed losses along the sensing fibre.
Distributed genetic algorithm
Genetic algorithms (GAs)^{70} are efficient search and optimisation techniques based on the principle of natural genetics evolution, aiming at finding optimum or suboptimum solutions to problems that are difficult to be algebraically solved. Standard GAs perform the search through a search space involving a population of individuals, each of which is randomly set following predefined initial conditions and represents a potential solution for the given problem. During the search process, the algorithm iteratively evaluates the fitness value of each individual and breeds new individuals by carrying out operations found in natural genetics. Once given termination conditions are satisfied, the searching process stops, and the individual exhibiting the best fitness value is output as the solution. The key idea of GAs is that the new generation of population bred after each iteration should involve individuals representing better solutions thanks to the use of classical operators inspired by the advancement in nature. These operators consist of: (1) selection, which selects the potential parents of offspring based on the evaluation of fitness value in such a way that better individuals are often selected for matching; (2) crossover, in which the genetic information from two parent individuals are exchanged to generate two offsprings; and (3) mutation, which creates new genetic information that does not exist in the parents. In order to expand the searching scope and to accelerate the search speed, DGAs^{66} that process multiple populations in parallel are often used. Furthermore, by periodically exchanging individuals among subpopulations via a migration operator, the premature convergence problem existing in conventional GAs can be greatly alleviated.
In this work a dedicatedly designed DGA is adapted to search for the optimal/suboptimal unipolar binary code sequence u(n) that can provide the smallest possible noise scaling factor Q (the largest possible coding gain G_{c}) for these given input parameters:

1.
the energy enhancement factor F_{E} bound by the physical constraints, as elaborated in Supplementary Notes 3 and 4;

2.
the value of m, which is a number between 3 and 4 according to the empirical results shown in Fig. 2b;

3.
the estimated envelope function f′(n) = u_{f}(n)/u(n) imposed by the EDFA saturation, which can be estimated from the selected m and EDFA specifications. Note that f′(n) is close to the N_{x}point downsampling of f(n), the latter being rigorously retrieved from the measured c_{f}(n).
The pseudocode for the designed DGA is sketched in Algorithm 1 and elaborated in the following, which basically consists of three sections: initialisation (line 1–2), genetic processing (line 4–10) and stopping criteria (line 3, 11–17).
Firstly, α subpopulations denoted as S_{i}(i = 1, 2, …, α) are defined, each of which contains β N_{u}bitlong allzero sequences (individuals) that are designated as u_{i,j}(i = 1, 2, …, α; j = 1, 2, …, β), where N_{u} = mF_{E}. Then the initialisation is performed by randomly replacing F_{E} ‘0’ bits by ‘1’ bits for each u_{i,j}, and the corresponding noise scaling factors Q_{i,j} are calculated according to Eq. (7). Then all subpopulations are processed in parallel by iteratively applying φ times the following genetic operators:
Selection: The selection operator updates a given subpopulation S_{i} by selecting individuals based on the criterion that sequences with smaller Q (as defined by our target) should have a higher chance to survive. To do so, a typical roulette wheel selection method^{70} is implemented, where the survival probability (i.e. the probability passing to the next generation) of each sequence u_{i,j} in a certain subpopulation S_{i} is expressed as:
from which it can be found that the smaller the Q_{i,j}, the larger the P(u_{i,j}). Using such a selection algorithm, the subpopulation is updated by carrying out the following selection steps for β rounds: (i) generate a random number P_{r} between 0 and 1, (ii) add P(u_{i,j}) to the partial sum \(\mathop {\sum}\nolimits_j {P(u_{i,j})} \), starting from the first sequence u_{i,1}of the given subpopulation till the condition \(\mathop {\sum}\nolimits_j {P(u_{i,j})} \ge P_r\) is satisfied, and (iii) select the first sequence that makes \(\mathop {\sum}\nolimits_j {P(u_{i,j})} \) exceeding P_{r} as the survival individual for the current round.
Crossover: The crossover operator is analogous to gene recombination, which produces offsprings by exchanging genetic material (i.e. alternating segments of code sequences) of the parents. The length of alternating segments is here set as \(N_u^c\) bits and onepoint crossover^{70} is applied, in which two sequences in a certain subpopulation are randomly chosen and their \(N_u^c\)bitlong segments starting from a random crossover point are swapped, generating two new offsprings. This process is repeated by \(\beta P_{c_i}\) times for a certain subpopulation S_{i}, where β is the number of individuals (sequences) and \(P_{c_i}\) is the probability of performing the crossover action.
Mutation: Mutation is an occasional tweak in the bit value of a sequence in order to maintain and introduce diversity in the subpopulation. This is usually applied with a low probability P_{m}, because if the probability is set very high, the GA would regress to an undesired random search. For the unipolar binary code sequence in our proposal, mutation is carried out by flipping the value of \(N_u^m\) randomly chosen bits of a sequence, which is repeated by \(\beta P_{m_i}\) times for a certain subpopulation S_{i}.
After finishing these φround iterations, the migration operation is subsequently performed to exchange the information among updated subpopulations. This is realised by transferring γ sequences with smaller Q in a source subpopulation to replace the same number of sequences with higher Q in a target subpopulation. Here each subpopulation S_{i} is in turn defined as the source of the neighbouring subpopulation S_{i+1}, for 1 ≤ i < α, while S_{α} is the source subpopulation of S_{1}. This way the information of ‘good’ sequences is spread among subpopulations; furthermore, the immigrants (i.e. γ sequences) would effectively change the fitness landscape thus avoiding the potential problem of premature convergence.
In parallel with the migration operation, the sequence with minimum Q (Q_{min}) among all subpopulations is selected as the best sequence u_{best} for the current generation (line 11 in Algorithm 1). The searching process is terminated when the converging condition is satisfied, i.e. when there has been no improvement of u_{best} for μ consecutive generations. This is indicated by a counter that is initialised to zero and counts the number of the generations for which there has been no improvement of u_{best}. As shown in lines 12–16 of the pseudocode, the counter increases every time a better u_{best} is not generated, which is however reset to zero if u_{best} is updated and \(\mathop {\sum}\nolimits_{n = 1}^{N_u} {f_d^\prime (n)u_{\rm{best}}} \left( n \right) \approx F_E\). The algorithm terminates when the counter value reaches μ and the u_{best} of the last generation is output as the final solution.
Algorithm 1
The distributed genetic algorithm.
Input: energy enhancement factor F_{E}, total bit number of a sequence N_{u} = mF_{E}, estimated envelop function f′, number of subpopulations α, number of individuals in each subpopulation β, crossover probability P_{c}, bit number of alternating segments \({N_u^c}\), mutation probability P_{m}, bit number of mutation \({N_u^m}\), migration interval φ, number of migrants γ, generation counter μ.
Output: the best sequence u_{best} and the minimum noise scaling factor Q_{min}.
1 Randomly initialise α × β sequences u, where α is the number of subpopulation and each subpopulation contains β sequences (individuals);
2 u_{f} = f′u, and evaluate the noise scaling factor Q for each u_{f} by Eq. (7);
3 while counter < μ do
4 for each subpopulation S_{i}(i = 1, 2, …, α) do φ times
5 S_{i} ← Selection (S_{i}, Q_{i,1}, Q_{i,2}, …, Q_{i,β});
6 S_{i} ← Crossover (S_{i}, \({P_{c_i}}\), \({N_u^c}\));
7 S_{i} ← Mutation (S_{i}, \({P_{m_i}}\), \({N_u^m}\));
8 Update the noise scaling factor Q_{i,1}, Q_{i,2}, …, Q_{i,β} based on u_{f} = f′u and Eq. (7);
9 end for
10 (S, Q) ← Migration (S, Q, γ);
11 (u_{best}, Q_{min}) ← (S, Q);
12 if u _{ best } is updated and \(\mathop {\sum}\nolimits_{{\boldsymbol{n}} = 1}^{{\boldsymbol{N}}_{\boldsymbol{u}}} {{\boldsymbol{f}}\prime ({\boldsymbol{n}}){\boldsymbol{u}}_{{\mathbf{best}}}\left( {\boldsymbol{n}} \right) \approx {\boldsymbol{F}}_{\boldsymbol{E}}} \) , then
13 counter ← 0;
14 else
15 counter ← counter + 1;
16 end if
17 end while
Note that the size of subpopulation, the parameters of genetic operations and the termination condition are all adjustable when using DGA. It has been verified experimentally that the proposed DGA with appropriate parameters is efficient in the search of optimal code sequence. Benefiting from the robustness of DGA, it is not necessary to tune all these parameters for a specific sequence length. All the numerical results in this paper are obtained with the single set of parameters shown in Table 2.
Code availability
The code of distributed genetic algorithm is deposited in Zenodo (https://doi.org/10.5281/zenodo.3909600).
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Acknowledgements
This study is supported by National Natural Science Foundation of China under Grant (61875018) and National Key R&D Programme of China under Grant (2017YFB0405500). M.A. Soto acknowledges the support from AC3E ANIDBasal Project FB0008. The authors thank Jianqi Hu from EPFL for the fruitful discussion on the coding optimisation.
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X.H. conceived the concept of deconvolution. Z.Y. and S.Z. developed the concept, established the theoretical model and designed the methodology. X.S. and Z.Y. developed the distributed genetic algorithm. X.S., Z.Y., X.H. and S.W. contributed to all experiments and analysis. M.A.S contributed to Raman experiments and background discussions. X.G. contributed to Brillouin noise analysis. X.S. and Z.Y. wrote the paper. M.A.S., X.H. and L.T. revised the paper. X.H., J.W. and L.T. supervised the project.
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Sun, X., Yang, Z., Hong, X. et al. Geneticoptimised aperiodic code for distributed optical fibre sensors. Nat Commun 11, 5774 (2020). https://doi.org/10.1038/s41467020192011
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