Enhanced Strain Measurement Sensitivity with Gold Nanoparticle-Doped Distributed Optical Fibre Sensing

. Nanoparticle-(NP-) doped optical fbres show the potential to increase the signal-to-noise ratio and thus the sensitivity of optical fbre strain detection for structural health monitoring. In this paper, our previous experimental/simulation study is extended to a design study for strain monitoring. 100 nm spherical gold NPs were randomly seeded in the optical fbre core to increase the intensity of backscattered light. Backscattered light spectra were obtained in diferent wavelength ranges around the infrared C-band and for diferent gauge lengths. Spectral shift values were obtained by cross-correlation of the spectra before and after strain change. Te results showed that the strain accuracy has a positive correlation with the relative spectral sensitivity and that the strain precision decreases with increasing noise. Based on the simulated results, a formula for the sensitivity of the NP-doped optical fbre sensor was obtained using an aerospace case study to provide realistic strain values. An improved method is proposed to increase the accuracy of strain detection based on increasing the relative spectral sensitivity, and the results showed that the error was reduced by about 50%, but at the expense of a reduced strain measurement range and more sensitivity to noise. Tese results contribute to the better application of NP-doped optical fbres for strain monitoring.


Introduction
Fibre optic sensors (FOS) ofer an attractive option for strain sensing, which can be used for structural health monitoring (SHM) decision-making.Structural damage will cause departures in the strain feld after loading [1].Te presence or position of damage can be obtained by analysing the strain information continuously.Compared with electrical strain gauges, FOS have the advantages of resistance to corrosion [2], immunity to electromagnetic interference [3], and their small size and low weight make them suitable to embed in composite materials [2].Tey can also measure temperature [4], chemical parameters (pH, relative humidity) [5], material degradation [6], vibration [7], and load [8].Because of these advantages, FOS has been used for SHM in felds such as aerospace engineering (for example, SHM in aerospace structures [9,10]) and civil engineering (for example, the monitoring of the railway infrastructures [11] and the pavement layers [12]).Te most commonly used optical fbre sensing methods for SHM include distributed fbre optic sensing [13] and fbre Bragg gratings sensing [14,15].Compared with fbre Bragg gratings, distributed fbre optic sensors (DFOS) have the advantage that they can obtain strain information along the length of the sensing fbres.One type of DFOS is based on the working principle of Rayleigh backscattering, where spectral wavelength shifts of the backscattered light in the optical fbre under strain are used to obtain the strain change values [13].Combined with the optical frequency domain refectometry (OFDR) method [16], this allows the distributed strain information to be demodulated.High spatial resolution distributed strain sensing based on Rayleigh scattering was developed by [13] with a strain resolution of 5 μϵ and a spatial resolution of 30 cm.It has since been developed to have a strain resolution of up to 1 μϵ with a spatial resolution of up to 1 cm [17] and has been developed for long-distance (more than 300 m) detection [18].Tere are plenty of applications of DFOS for SHM, for example, damage detection for a carbon fbre reinforced plastic (CFRP) beam and a composite wing [19] and the state sensing of a composite winglet structure [20].
Te strain measurement performance of a DFOS directly determines the fnal performance of SHM, and highperformance sensing is benefcial to SHM [21].Some drawbacks restrict the application of DFOS for some applications.Te durability of the optical fbre sensor in engineering has drawn researchers' attention [22].Te strain transfer between the materials and the sensing fbre should be considered for accurate strain measurement in engineering [23].Another limitation which restricts DFOS strain measurement based on Rayleigh backscattering has drawn many researchers' attention in recent years is that this method uses the backscattered light in the core of the optical fbres, while the Rayleigh scattering in the core of commercial optical fbres is low [24].For a commercial optical fbre, the backscattered light signal is as low as − 100 dB/mm [25].By increasing the backscattered light in the core of the optical fbres, the signal-to-noise ratio (SNR) may increase, allowing the precision of the strain detection to increase.Some previous research shows improvements in the strain/temperature measurement based on increasing the backscattered light signals.Loranger et al. increased the magnitude of backscattered light by 20 dB by ultraviolet laser exposure (213 nm wavelength) of the core of the optical fbres [24], and the strain/temperature measurement noise level decreased.Parent et al. achieved about 37 dB backscattered light enhancement with the ultraviolet laser exposure of the core of optical fbres and the accuracy of the shape sensing of the surgical needles increased by 47% [26].Doping nanoparticles (NPs) into the core of the optical fbre is a direct approach to increasing the backscattered light.Te engineering features of NP-doped optical fbres have been investigated for manufacturing, signal attenuation, and light enhancement with diferent types of NPs (MgO-based NPs [27] and Ca-based NPs [28,29]).Gold NPs are competitive contrast agents that can increase backscattered light [30].If gold NP-doped optical fbre is used in strain detection for SHM, a higher SNR signal may improve the precision of the strain measurement for SHM which is benefcial for critical locations.For example, critical locations include the areas close to the rivet holes in the aluminium fuselage where small cracks may form [31]. Tese small cracks will induce a small localised strain change compared to a healthy structure, so a high strain sensitivity is needed for detection.
In our previous work, gold NP-doped optical fbres were investigated by experiments and simulations.To investigate the backscattered light enhancement in an optical laboratory, a method of doping liquid containing gold NPs onto the optical fbre end tips was proposed to analyse the intensity of backscattered light signal scattered by the NPs [32] and analyse the spectral shift of the NPdoped optical fbres under strain experimentally [33].By simulation, the optimised light enhancement at 1550 nm wavelength was obtained (about 64 dB) [30].With NPdoping, the intensity of the backscattered light in the optical fbre will increase to overcome the drawback of low backscattering intensity of commercial optical fbres.A comparison between diferent methods of backscattered light enhancement in the core of the optical fbres is listed in Table 1, and gold NP-doped distributed fbre sensing shows the advantages of light enhancement.Te NP-doped optical fbres' backscattered light spectra difer from the commercial signal mode optical fbres or fbre Bragg gratings.Te new spectral characteristics of NP-doped optical fbre under strain were obtained in previous work [34].
Te previous research showed that the dramatically increased backscattered light signal benefts the strain/ temperature measurement [28][29][30].Te sensitivity is the minimum signal that can be detected and the research on sensitivity improvement is an important topic.Loranger et al. [24] performed research on the sensitivity improvement of the distributed fbre optic sensing for different gauge lengths with backscattered light enhancement method for some cases by measuring the root mean square noise.However, studying the performance of strain measurement under diferent noise levels for diferent cases experimentally is a challenge and the experimental results only showed that the signal-to-noise ratio increases from the engineering aspect.Te theoretical performance of the NP-doped FOS for strain monitoring still needs to be investigated.In addition, the backscattered light spectra of NP-doped optical fbre are diferent from commercial optical fbres.In this case, the traditional cross-correlation method to demodulate the spectral shift under strain should be developed for a more accurate strain/temperature measurement.
In this paper, strain acquisition of gold NP-doped distributed fbre optic sensing will frst be investigated with the cross-correlation method.Te errors caused using the crosscorrelation method for NP-doped optical fbre will be evaluated for diferent random-seeded optical fbre gauge lengths without noise and for the specifc random-seeded optical fbre gauge lengths under diferent noise levels.Gaussian noise is assumed and applied to the photoelectric conversion signals.By linear ftting, the sensitivity of the NPdoped optical fbre strain gauges can be obtained.To improve the accuracy of strain detection with the crosscorrelation method for NP-doped optical fbre sensors, a new method is proposed to increase the accuracy of strain detection.
Te paper is organized as follows.Te frst section is the Introduction.Te experimental setup, the principle of the measurements, and the proposed method for increasing the accuracy of spectral shift detection are illustrated in the Methodology section.Te third section shows the simulation results for diferent random-seeded optical fbre gauge lengths without noise and for the specifc random-seeded optical fbre gauge lengths under diferent noise levels.Analysis from the simulation and a strain measurement application (an aerospace case study) will be shown and discussed in the Analysis and Discussion section to present the sensitivity of the NP-doped optical fbre sensors and to make a comparison with the traditionally used FOS.Ten, the results of the strain acquisition with the new proposed method will be discussed.Te ffth section is the Conclusions section.

Methodology
2.1.Experimental Setup and the Principle.Figure 1(a) shows a simplifed diagram of the optical fbre sensing system based on backscattering by the NPs. Figure 1(b) shows the structure of the strain measurement system monitoring strain distribution in a specimen under load with an optical fbre.In Figure 1(a), the optical fbre sensing system consists of a wavelength-tunable laser, optical fbre couplers, an optical fbre circulator, optical fbres, a sensing fbre containing gold NPs, a photodetector, and an optical trap.Te light signals detected by the photodetector are processed by the computer which is not shown in fgure.
Light emitted from the tunable laser is split by an optical coupler into sensing and reference paths.Te gold NPs are randomly distributed in the sensing fbre core volume.Light is backscattered by the NPs and recoupled into the optical fbre in the backward direction.An optical trap at the end of the sensing fbre is used to reduce the refection at the optical fbre end tip, which can be a refractive index (RI) matching liquid or a no-core optical fbre.Te backscattered light passes through the optical circulator and then interferes with light from the reference arm at the optical fbre coupler.Optical beat signals obtained at the photodetector are demodulated by Fourier transform to recover the intensity and phase of the backscattered light in each gauge length.
Te beat signals can be expressed as follows [38,39]: where r i is the refection coefcient at position z i of the sensing fbre and the squared magnitude |r i | 2 is the refectance.Te spatial distances between the i th and the (i + 1) th detection points (spatial resolution) are determined by the efective RI of the optical fbre and the tuning wavelength range is ∆z � λ s λ f /2n∆λ, where λ s is the starting wavelength of the tunable laser and λ f is the fnal wavelength, ∆λ is the wavelength tuning range, n eff is the efective RI of the optical fbre [40] and k is the wavenumber.For a tunable laser source with a starting wavelength at 1540 nm and a fnal wavelength at 1560 nm, the spatial resolution is about 41 nm when the RI is 1.45.If the gauge length is set at 1 mm, there are about 24 detection points in the gauge length.It can be seen from equation ( 1) that PD 2 contains information about the position (z i ) and the wavenumber (k) and it can be transformed by Fourier transform.Because the laser is linear tuning, the beat signals are proportional to the positions of z i [41].Te intensity and phase of the scattered light spectra within the gauge length can be obtained by square windowing the positions within the gauge length and using an inverse Fourier transform.Ten, the spectral shift can be obtained using cross-correlation between the reference spectrum and the spectrum after the strain is applied [42].

Te Method of the Spectral Shift Detection by Cross-Correlation Method with Gold NP-Doped Optical Fibre.
Te accuracy of the cross-correlation was evaluated by comparing the theoretical spectral shift under strain and the spectral shift result from the cross-correlation method.Te central wavelength of the tunable laser was set at 1550 nm.Te wavelength tuning ranges were set at 10 nm, 20 nm, and 40 nm, which correspond to the wavelength ranges 1545 nm-1555 nm, 1540 nm-1560 nm, and 1530 nm-1570 nm, respectively.Te simulated spectra were in the wavelength domain with a high spectral resolution of 1 pm in order to accurately reconstruct the spectra.In this case, the spectral shift (R xy ) was calculated by the cross-correlation method in the wavelength domain as follows: where n � − (N − 1), − (N − 2), . . ., N − 2, N − 1. N is the number of sampling points in wavelength range.Te sampling points were mapped to a uniform scale in the wavelength domain.Te resolution was set at 1 pm.y and x correspond to the reference spectrum and the spectrum under strain, respectively.y and x were normalised to the range of 0 to 1. Te strain was applied to the sensing fbre by a controlled load to apply force on the optical fbres.In the experiment in this paper, a controlled tensile was applied to the specimen containing an optical sensing fbre to change the strain of the sensing fbre.In this case, the relative positions of the gold NPs in the sensing fbre will change and the RI of the optical fbre will change also.Te backscattered spectra from the NPs in the gauge length were modelled by a single scattering model which means the light will be backscattered once in the propagation direction and can be expressed as follows [34]: Structural Control and Health Monitoring where λ NP ′ is the spectrum under strain and λ NP is the reference spectrum.η m � − (p 12 − ](p 11 + p 12 ))n 2 /2 which is the parameter caused by the RI change under strain [43].When ] � 0.17 [44], p 11 � 0.113, p 12 � 0.252 [44], and n � 1.45 for the optical fbre, η m � − 0.1997.Terefore, the theoretical spectral shift under strain is 1.240465 pm/μϵ which was set as the true value used to compare with the simulated results.
From the previous work, low-concentration NP-doping shows similar spectral characteristics in the wavelength and wave number domains [34] for diferent sizes of NPs.To simplify the simulation, only 100 NPs (100 nm diameter) were randomly seeded in the gauge lengths for a low concentration of NP doping.Te amplitudes of the scattered light were calculated with Mie theory.Once the NPs have been seeded, the spectral intensity can be expressed by the square of the accumulation of the electric feld caused by the scattering from the NPs as follows [34]: where E 0 is the electrical feld of the incident light, is the refection coefcient of the NP at position of l i in the backward direction in the optical fbre which is calculated with Mie theory, f is the volume ratio of gold NP in the optical fbre, n m is the RI of the optical fbre, R is the real symbol, S(0) is the forward scattering amplitude, λ is the wavelength of the incident light, x is the size parameter of the gold NP, j is the imaginary unit, and ε represents axial strain.Te simulations of the backscattered light spectra were performed analytically and the simulation procedure is shown in Figure 2.  4), where the Mie theory and numerical aperture were applied to the simulations.Te spectral shift values and then the strain values were calculated with the cross-correlation method.To analyse the relative spectral sensitivity, the backscattered light spectra can be transferred to spatial frequency (1/λ) by Fourier transform to evaluate the fuctuation within 1 nm.In order to obtain this fuctuation, a threshold was set at 1/e of the maximum value in the spatial frequency domain.Te frst spatial frequency value over the threshold from half of its maximum spatial frequency to the low spatial frequency was chosen as its fuctuation number was within 1 nm.Note: the spectra were normalised to the range 0-1 and were then subtracted from the mean values of the spectra.
Te parameters used for the simulations are shown in Table 2.In the simulation, the geometry of the optical fbres was cylinders.Te gauge lengths were chosen to be integer multiples of the grating period of the fbre Bragg gratings for a wavelength at 1550 nm for a separate study.Te reasons for choosing the strain range (− 3000 μϵ to 3000 μϵ) and strain interval (100 μϵ) are that it is a relatively wide strain range to cover the strain values in the loading test for the specimen in this paper and the strain interval's choosing has balanced the time-consuming and resources consuming for the computing.Te number of the NPs in the core of the optical fbre was set at 100.Te reason for choosing 100 NPs is that 1: it is a very low concentration and it meets the single scattering   Structural Control and Health Monitoring assumption (l < 1/c/C ext , where l here is the gauge length, c is the concentration of the NPs and C ext is the extinction crosssection for the NP which can be calculated by Mie theory) for the set gauge lengths; 2: the spectral characteristics show similarities when it meets the single scattering requirement according the results from the research [34].In this case, the generated spectra are general and can also be valid for other concentration cases.Te results also apply to other sizes of NPs, even though the sizes of the NPs were set at 100 nm in this paper.

A New Method for Increasing the Accuracy of Spectral Shift
Detection.It was found that the accuracy of the crosscorrelation relies on the characteristic spectral peaks.For the cases of shorter gauge lengths (see Simulation Results section), the spatial frequency is less than the cases of longer gauge lengths.It means that the characteristic spectral peaks of the shorter gauge lengths are less than the cases of the longer gauge lengths, which will lead to the lower accuracy of spectral shift monitoring for short gauge lengths with the cross-correlation method.Based on this, in this section, a method to increase the accuracy of spectral shift monitoring based on increasing the spatial frequency is proposed.Te method to increase the characteristic spectral peaks is using partial spectral inversion.Te spectra processing procedures are as follows: Firstly, the original spectra were normalised to the range 0-1 (x and y).Secondly, the direct current part of the normalised spectra was deduced by subtracting their mean values (x(m) − x and y(n + m) − y).Ten, the values of the spectra are around zero.Tirdly, by spectra inversion operation of the spectra below zero (|x(m) − x|) and |y(n + m) − y|)), the absolute values of the spectra were obtained.Fourthly, the direct current part of the normalised spectra was deduced by subtracting their mean values (|x(m) − x| and |y(n + m) − y|).Te fnal step is using the cross-correlation method for the processed reference spectrum and the spectrum under strain to obtain the spectral shift.By a spectra inversion operation, additional high-frequency information was added to the original spectra.
Te proposed method can be expressed as follows:  where N is the number of the wavelengths used for calculation, x and y refer to the normalised reference spectrum and the spectrum under strain respectively.x and y are the mean values of the reference spectrum and the spectrum under strain, respectively.A case for the spectra generated by 100 nm size gold NPs randomly distributed in the optical fbre is shown in Figure 3. Te gauge length was set at 0.3207 cm.Te spectral range was set from 1545 nm to 1555 nm.Te wavelength resolution was set at 1 pm.
In Figure 3(a), the reference spectrum is shown by blue lines and the spectrum under strain at − 1200 μϵ is shown by black lines.Te spectrum under strain shows a left spectral shift when it is compared with the reference spectrum.Te black arrow shows the direction of the spectral shift of one of its major characteristic spectral peaks.Te characteristics of the spectral shift under strain of the NP-doped optical fbre show similar spectral shift characteristics of fbre Bragg gratings but the characteristic peaks are quite diferent from the Bragg peaks.In order to show the characteristics of the fuctuation of the peaks, the Fourier transform is applied to the backscattered light spectra to show the number of fuctuations of the spectra within one wavelength (relative spectral sensitivity).Te results of the spectra shown in Figure 3(a) after the Fourier transform are shown in Figure 3(b).Half of the sampling frequency is 500/nm, which is much larger than the 3 dB frequency of the spatial frequency.In this case, the sampling rate is two times larger than the Nyquist frequency.Terefore, the spectral information can be recovered with this resolution.Te results after Fourier transforms in Figure 3(b) have been reduced by subtracting the direct current component in Figure 3(a).Te traditional cross-correlation method is used for the spectral shift measurement with the spectra in Figure 3(a).
Figure 3(c) shows the processed spectra with the proposed method.Figure 3(d) shows the intensity information after the Fourier transform.Compared with Figure 3(a), with the new method, the characteristic peaks in Figure 3(a) increase.In this case, the relative spectral sensitivity increases which can be seen in Figure 3(d).Tere are more high-frequency components shown in Figure 3(d) than the results in Figure 3(b) for both the reference spectrum and the spectrum under strain.Te corresponding spatial frequency at the threshold (1/e) may be diferent for the reference spectrum and the spectrum under strain because some of the major characteristic peaks may move outside the measured spectral range.In addition, the obtained spatial frequencies after the Fourier transform are discrete values.Terefore, the end point (the nearest spatial frequency at which the intensity of the result after the Fourier transform is above the threshold) is used for the reference spectrum to compare the fuctuation of the spectra used in traditional crosscorrelation method and the new method.With the new method, the spatial frequency at the end point increases when comparing Figure 3(d) (2.4 nm − 1 ) and Figure 3(b) (1.6 nm − 1 ), which will be benefcial for a more accurate strain measurement (in Results of the newly proposed method in the Analysis and Discussion section).
In this work, the performance of the strain monitoring of signal-enhanced distributed fbre optical sensing based on backscattering by doping gold NPs will be investigated by simulations with a single scattering model in order to improve the sensing performance.Te simulation results of the NP-doped optical fbre sensors will be shown in the next section.

Simulation Results
Te results of the cases of the accuracy of the spectral shift acquisition based on the cross-correlation method will be shown.Te frst three cases were generated by 100 times random seeding NPs without noise for diferent gauge lengths.Te following three cases were generated with the same selected random seeding NPs distribution with different noise levels and diferent gauge lengths to assess the accuracy with the cross-correlation method used for NPdoped optical fbre sensors.

Spectral Analysis with 100 Random Seedings without
Noise. Figure 4 shows the strain errors between the simulated and theoretical results.Te simulated results were obtained with the cross-correlation method to calculate spectral shifts under strain for spectral detection ranges 10 nm, 20 nm, and 40 nm with the central wavelength of 1550 nm in Figures 4(a)-4(c), respectively.Te true values (theoretical results) were obtained by using the theoretical response of the spectral shift (1.240465 pm/μϵ) multiplied by the strain values.Te strain range was set from − 3000 μϵ to 3000 μϵ and the strain interval was set at 100 μϵ.Te gauge lengths were set at 0.1069 cm, 0.2138 cm, 0.3207 cm, 0.4276 cm, 0.5345 cm, 0.6414 cm, 0.7483 cm, 0.8552 cm, 0.9621 cm, and 1.0690 cm, respectively.
To show the ratios of the deviation between the calculated mean values and the true values, the relative error is defned as follows: Te relative errors between mean values and the true values are shown in Figures 4(a-1)-4(c-1).Tey were obtained by averaging the 100 cases of the obtained errors.Because equation ( 6) cannot be used for the cases for strain at 0, the results close to strain at 0 are not available.
In the small gauge length ranges, as seen from Figure 4(a-1), the calculated mean spectral shifts were not sufcient to reach the true spectral shifts, especially for larger strain values.A reason for this may be that a large strain causes a large spectral shift and a larger spectral shift causes a higher percentage of the spectrum moving out of the spectral detection range and being replaced by a new spectrum and the new spectrum has no correlation with any regions of the original spectrum.Te high errors shown in Figure 4(a-1) represent the failure of strain acquisition with the cross-correlation method.To show the degree of dispersion of the 100 randomly seeding cases, the errors calculated with the minimum data and the maximum data in the 100 cases for each gauge length and strain were compared with the true values shown in Figures 4(a-2) and 4(a-3), respectively.A threshold of 20% was set for both fgures to show the regions where the errors are above 20% and more details about the regions where the errors are below 20%.It is interesting to see that signifcant errors only exist in positive strain regions in Figure 4(a-2) for the calculated minimum errors and only exist in negative strain regions in Figure 4(a-3) for the calculated maximum errors.Tis indicates that some cases failed when the cross-correlation method was used.Figure 4(a-4) shows the standard deviations of strain errors.Te white line shows a boundary around the threshold in Figure 4(a-4) and the white line passes through (0 μϵ, 0 cm) and (2300 μϵ, 1.0690 cm).Te boundary lines shown in Figure 4(b-4) pass through (0 μϵ, 0 cm) and (− 2000 μϵ, 1.0690 cm).Te two white lines were also drawn in Figure 4(a-4) and similar boundaries are shown when the threshold was set at 0.5%.Te standard deviations are quite small but the errors calculated with the maximum results and minimum results shown in Figures 4(a-2) and 4(a-3) are large, which also indicates a failure of the spectral shift acquisitions with the crosscorrelation method and indicates that the failure cases have a low proportion for the 100 randomly cases.
Figures 4(b-1)-4(b-3) show errors between the mean values and the theoretical values, the errors between the minimum calculated results and the theoretical values, and the errors between the maximum calculated results and the theoretical values, respectively.Te errors between the mean values and the theoretical values shown in Figure 4(b-1) are small compared with the results in Figure 4(b-1).Only the regions close to the smallest gauge lengths for some large strain values show high error values.Because the relative errors are quite small, the threshold was set at 2% for the results shown in Figures 4(b-2) and 4(b-3).With the 2% error threshold, the regions of the errors above the threshold are similar to the regions of the errors above the threshold of 20% shown in Figures 4(a-2) and 4(a-3).Figure 4(b-4) shows the standard deviations of strain errors.In Figure 4(b-4), the errors are reduced to lower values compared with Figure 4(a-4), especially for the regions for larger strain and small gauge length.However, errors still exist in regions with a small strain and a small gauge length.Figure is not symmetric as can be seen from the dark blue and light blue zones in Figure 4(b-4).Te reason for this is unclear and needs to be investigated further.
Te results for the 40 nm spectral detection range (1530 nm-1570 nm) case are shown in Figure 4(c).Figures 4(c-1)-4(c-4) show the errors between the mean  Structural Control and Health Monitoring values and the theoretical values, the errors between the minimum calculated results and the theoretical values, the errors between the maximum calculated results and the theoretical values and the standard deviations of strain errors, respectively.
Te errors between the mean values and the theoretical values in Figure 4(c-1) are smaller than the results shown in Figures 4(a-1) and 4(b-1).Only the regions close to the smallest gauge lengths show high error values.Te threshold was set at 2% for the results shown in Figures 4(c-2) and 4(c-3) which is the same as the threshold in Figures 4(b-2) and 4(b-3).Te threshold set in Figure 3(c-4) was 0.5%.With the same thresholds, it can be seen that the errors decrease when the spectral ranges increase.
Te above results show the randomly positioned NP cases and the corresponding statistical results for diferent spectral ranges without noise.In the following part, diferent levels of random intensity noise will be added to the spectral signals obtained from a random spatial distribution in the same gauge lengths, strain ranges and spectral ranges.Te failure caused by the cross-correlation was avoided in this selected NP relative spatial distribution.

Spectral Analysis with a Specifc Random Seeding with
Noise. Figure 5(a) shows the direct errors between the mean values of the spectral shifts calculated with the cross-correlation method with diferent noise levels.Te strain values can be obtained by using the theoretical response of the spectral shift.Te direct error was defned as follows: Te mean values were the averaged results of 100 spectra superposed with randomly generated noise.Te noise is assumed to follow a Gaussian distribution with a mean value of 0 and a standard deviation of σ n .Te noise level in the simulation was defned as follows: where max I s   represents the maximum value of the original backscattered light spectrum without noise in the gauge lengths.It is assumed that σ n is independent of the intensity of the signal.Figures 5(a-1)-5(a-3) show the errors between -3000 3000 -1500 1500 0 Strain (µ∈)  (Te results for noise levels of 10%, 20%, 30%, 40%, and 50% are not shown because noise shows no infuence on the mean values).Te original spectrum without noise was normalised to 0-1 and then noise with noise level N was added to the original spectrum.By setting the noise level, spectra with diferent noise levels can be simulated.In this case, the approximated SNR ratio can be expressed as follows: or as SNR L ratio in dB as follows: Figure 5(a) shows the errors between the calculated values and the true values for 10 nm, 20 nm, and 40 nm spectral ranges, respectively.In Figure 5(a-1), the errors  -d) correspond to results under noise levels of 10%, 30%, and 50%, respectively [45].
between the calculated values and the true values become notable when the gauge lengths are small.Tere is a noticeable error around the point (− 1200 μϵ, 0.3207 cm).Te reason for this noticeable error can be illustrated with the spectra shown in Figure 3(a).Figure 3(a) shows the corresponding spectrum for the strain at − 1200 μϵ and for the gauge length set at 0.3207 cm.Under the strain value of − 1200 μϵ, the original spectrum in blue has a blue shift.Te arrow shows the spectral shift under this strain value and the spectrum under this strain value was in green.As shown in Figure 3(a), there is a distinguishable peak near the arrow and the intensity of the peak is much larger than the other peak values shown in the spectral range.Terefore, the shift of this peak has a high infuence on calculating the spectral shift under strain with the cross-correlation method.For the strain value at − 1200 μϵ, the peak shifts out of the spectral range which is a reason for the high error near this point shown in Figure 3(a).Tis result indicates that the accuracy of obtaining the spectral shift not only depends on the gauge lengths and the strain ranges but the accuracy also depends on the characteristics of the original spectrum.Figures 5(a-2) and 5(a-3) show smaller errors than errors in Figure 5(a-1), but the characteristics of the original spectrum depend on the distribution of the NPs.For example, in Figure 5(a-2) a remarkable error occurs at negative strain and the shortest gauge length for calculation.Te positions of the high error points depend on the characteristics of the original spectrum.By adjusting the spectral range, the high error points can be moved.Figures 5(b)-5(d) show the standard deviations of the spectral shift obtained with the cross-correlation method with the same random NP seeding distribution in the gauge lengths with the 10 nm, 20 nm, and 40 nm spectral range under diferent noise levels (10%, 30%, and 50%), respectively.As the noise level increases, the standard deviations increase especially for the short gauge lengths.For a specifc noise level, the standard deviations of the results caused by the noise are similar for diferent strain values.Te standard deviation shows a positive relevant relationship with the noise level.

Analysis and Discussion
In the previous subsections for diferent spectral ranges with random seedings, the results for the accuracy of spectral shifts demodulation with cross-correlation method were shown.Te value of spectral shift times the theoretical responsivity of strain (about 1.24 pm/μϵ) is the obtained strain results.By increasing the detected spectral range, the accuracy of the strain acquisition increases with the crosscorrelation method.Although these results show the tendency of some relationships, the mathematical relationship between parameters (for example, spectral range, and gauge length) is unclear.Te failure of the strain acquisition shown in the results of subsections of spectral range 1545 nm to 1555 nm with random seedings, spectral range 1540 nm to 1560 nm with random seedings and spectral range 1530 nm to 1570 nm with random seedings causes difculty to obtain a formula to show the relationships, so the results from the subsections of spectral ranges 1545 nm to 1555 nm, 1540 nm to 1560 nm, and 1530 nm to 1570 nm with the specifc random seedings are used for further analysis since these results show the strain acquisition without failure.
In this section, further analysis of the parameters (spectral range, gauge length, and signal-to-noise ratio) will be investigated in order to obtain a formula for evaluating the sensitivity of the NP-doped optical fbre sensors.Tis is followed by an aerospace case study of real strain data obtained by LUNA system (ODiSI-B) with commercial optical fbres which is used as a comparison with the simulated results from NP-doped optical fbre sensors.Finally, the results of the proposed new spectral shift demodulation method are shown and discussed.

Sensitivity of the NP-Doped Optical Fibre Sensors.
Figure 6 shows the relationship between the reciprocal of spatial frequency and the modifed standard deviation of the cross-correlation method.Te horizontal axis is the reciprocal of spatial frequency (R) and the vertical axis is the modifed standard deviation (M).Tey show an approximate linear relationship, which will be easier for linear or polynomial ftting.
Tere are 15 groups of data shown in Figure 6.Te datasets with the spectral ranges 10 nm, 20 nm and 40 nm are shown in red, blue and green respectively.Te dataset with noise levels of 10%, 20%, 30%, 40%, and 50% are shown with a pentagram, asterisk, plus sign, circle and a triangle, respectively.Te reason that there is no data shown in Figure 6 with a noise level of 0 is that the signal-to-noise ratio is fnite when the noise is 0 and the modifed standard deviation was defned as M � D/N.D represents the standard deviation of the obtained spectral shift and N is the noise level.In this case, the standard deviation is always 0 for the noise level of 0. Te reason for using modifed standard deviation rather than standard deviation is that standard deviation cannot be distinguished when the noise level is small.By dividing the noise level, the modifed standard deviation can be shown in an approximated linear function, although the deviation may be large for small noise levels.
By a linear ftting, a relationship between the modifed standard deviation and the reciprocal of spatial frequency can be found (R 2 � 0.8993): Te reciprocal of spatial frequency (R) is defned as R � 1/F.F is the spatial frequency of the spectrum after the Fourier transform which meets the threshold of 1/e which is defned as the frst spatial frequency point that is over the threshold of 1/e and this frst point was chosen from the high frequency, beginning with half of the total spatial frequency after the Fourier transform to lower frequencies.F � P/W, where P is the spatial frequency in the spectral range and W is the wavelength range with the unit of nm.Along with the defnition of SNR as SNR L � 1/N 2 , the relationship between the noise level, the spectral range, the gauge length, the standard deviation of the calculated spectral shift with the cross-correlation method can be expressed as follows: Equation ( 12) may be used for fast evaluation of the infuence of the noise level, from the spectral range for the detection, and the intended gauge lengths.
It has been shown that there is an approximately linear relationship between the spatial frequency and the gauge length.According to the defnition of the spatial frequency in this paper, the mean spatial frequency of the cases shown in Figure 5 for the noise level of 0% is shown in Figure 7. Te response of the spatial frequency to the gauge length is about 10 nm/cm.Te red line shown in Figure 7 shows a linear function with a slope of 10 nm/cm and passing through the origin.
Ten, with this approximated slope value, equation ( 12) can be adjusted as (R 2 � 0.8602): where L is the gauge length in centimetres.It can be seen from, equation ( 13) that the standard deviation is proportional to the reciprocal of the square root of SNR.By increasing the SNR, the precision of strain detection will be improved.
If the sensitivity is defned as the value of the minimum input strain when the signal-to-noise ratio of strain (SNR ε ) equals 1, then a sensitivity (S) is equal to the deviation D in noise signals as follows: when the responsivity is set at 1.24 pm/μϵ.If the photon noise obeys a Poisson distribution and the light intensity is proportional to the gauge lengths, then the sensitivity formula can be expressed as where SNR L 0 is the signal-to-noise ratio of the spectra for the calibration gauge length of L 0 .

Strain Measurement Application (An Aerospace Case).
SHM integrates sensors within structures to record damage evolution and provides information for structural integrity analysis.Te integrated sensor can perform real-time monitoring of structures, and that may reduce nondestructive evaluation frequency and thus decrease maintenance cost as well as increase structural life safety, which is vital in areas such as aerospace engineering [46].Sensors can  Figure 6: Te relationship between the reciprocal of spatial frequency and the modifed standard deviation of the crosscorrelation method.Te noise levels are 0%, 10%, 20%, 30%, 40%, and 50%.Te spectral wavelength ranges are 10 nm, 20 nm, and 40 nm.Te red line shows the linear ftting results [45].
Structural Control and Health Monitoring be mounted on structures externally or embedded within structures [47].Both internal and external sensors can be utilized for strain measurement.
Aluminium is a material widely used in aerospace engineering.FOS was integrated into an aluminium alloy part to analyse the infuence of the position of the optical fbre sensor in a capillary [48].Te material of the aluminium alloy chosen was AlSi10Mg because AlSi10Mg has the advantage of good strength and low weight properties.It is often used for additive manufacturing in aircraft [49] and the specimen was manufactured with selective laser melting.Te embedded optical fbre sensor was used to monitor the strain information to record damage evolution and to provide information for structural integrity analysis.
In order to make a comparison between the simulated results of the NP-doped optical fbre and commercial optical fbres, an experiment of an aerospace case was implemented to obtain real strain data.A set of experimental strain data was obtained from a commercial optical fbre sensor which was interrogated by the LUNA system (ODISI-B) at DASML of TU Delft.
Figure 8 shows a photo of the experimental setup used for monitoring the crack propagation with a four-point bending test.Te aluminium alloy part was installed on a fatigue test machine (MTS servo-hydraulic test frame).Te optical fbre sensor was embedded into the capillary of the aluminium alloy with injected adhesives.An initial notch was made at the middle of the bottom of the specimen.By loading of 8 kN employed on the specimen, the crack initiated and propagated.More detail of the structure of the specimen is shown in Figure 9.
Figure 9 shows the structure of the specimen on the fourpoint bending fatigue test machine.Tere is a cylindrical capillary with a 3 mm diameter (between the white dashed lines).Te centre of the capillary is about 16 mm to the bottom of the specimen.Force was applied on the top of the specimen and two holders supported the specimen under loading.Te initial notch position is shown in Figure 9.
Generally, when the crack is generated close to the optical fbre sensor, the local stress will cause a local strain change.By analysing the local strain distribution, the crack could be monitored.Figure 10 shows the experimental results of strain distributions in the optical sensing fbre for diferent crack lengths (crack lengths: 2 mm, 4 mm, 8 mm in blue, yellow, and green, respectively).Te data were averaged from 100 testing results.Te crack length was defned as the distance between the end of the crack to the bottom of the specimen.It can be seen that there is a peak in the middle of the sensing fbre and the peak increases when the crack propagates from 2 mm in length to 8 mm in length under load.When the crack length is small, for example, 2 mm, the shape change from the original uniformed strain is relatively small and it is comparable to the standard deviations (red bars).Tis makes it difcult to detect small cracks from the strain distribution pattern.
However, this drawback will be overcome when the backscattered light enhancement methods are used.Te sensitivity of the strain sensing will increase when the backscattered light increases.By ultraviolet (UV) light exposure, the backscattered light in single mode fbre − 28 (SMF-28) will increase by 20 dB [24].With nanoparticle doping, more than 40 dB backscattered light increase has been achieved [27,29].Te obtained standard deviations of strain from the experiment are about 11 μϵ from the commercial optical fbre.From equation ( 14), the strain standard deviations obtained from NP-doped optical fbres can be lower than that obtained from commercial single-mode optical fbres when the SNR L 0 is above 19 for a gauge length of 0.13 cm (L 0 ) and it will be easy to achieve a SNR L 0 of more than 19 with these backscattered light enhancement methods because of the dramatically increased backscattered light intensity.
Figure 11 shows a comparison between the simulated results and the experimental results from this work and the data from Loranger et al.'s paper [24].Te black star shown in Figure 11 is the experimental result from this work, which was obtained based on the experimental data for the standard deviation (11 μϵ) obtained from strain monitoring in Figure 10.Te calibrated calculated sensitivity curve is shown in a blue line using this experimental data (SNR L 0 � 19 for 0.13 cm gauge length (L 0 )).Te formula of the curve is S � 0.3919/L 3/2 + 0.9513/L 1/2 .Te blue circles represent the sample points that were used as gauge lengths in previous sections which are 0.1069 cm, 0.2138 cm, 0.3207 cm, 0.4276 cm, 0.5345 cm, 0.6414 cm, 0.7483 cm, 0.8552 cm, 0.9621 cm, and 1.0690 cm, respectively.
Te experimental data from Loranger et al.'s paper [24] are used for comparison.Loranger et al. obtained temperature measurements with the distributed fbre optic sensing for diferent gauge lengths which are 0.05 cm, 0.1 cm, 0.2 cm, 0.5 cm, 1.0 cm, 2.0 cm, and 10.0 cm, respectively.Te calibration factor for strain and temperature used was 8.32 μϵ/ °C [24].With this calibration factor, the sensitivities for strain measurement were obtained for these gauge lengths (shown in the red triangles).Compared with the experimental data, the sensitivity curve (in blue line) shows the same tendency and matches the experimental data from Loranger et al.'s paper [24].

Results of the New Proposed Method.
Te errors of the obtained spectral shifts are related to the spectral range as shown before (see Figure 5(a)) and the spatial frequency is proportional to the spectral range [34].Terefore, increasing the spatial frequency from the original spectrum may increase the acquisition of the accuracy of the spectral shift and then increase the accuracy of the strain measurement.Te new method was proposed based on these fndings.
Figure 12 shows the results of the proposed method with the comparison from the cross-correlation method.Te spectral range used was from 1540 nm to 1560 nm in Figure 12.Te strain was set at 500 μϵ.Te gauge lengths were set at 0.1069 cm, 0.2138 cm, 0.3207 cm, 0.4276 cm, 0.5345 cm, 0.6414 cm, 0.7483 cm, 0.8552 cm, 0.9621 cm, and 1.0690 cm.
In order to show the accuracy improvement with the proposed method compared with the traditional crosscorrelation method, the error ratio was used in Figure 12, which is defned as the ratio of the relative error between the spectral shift calculated by the proposed method and the theoretical value to the relative error between the spectral shift calculated by the cross-correlation method and the theoretical value.
Figures 12(a) and 12(b) show the relative errors between the results calculated with the cross-correlation method and theoretical values in blue lines with circles and the relative errors between the results calculated with the proposed method and theoretical values in red lines with stars under noise levels of 0% and 10%, respectively.Te errors decrease to half of their values for short gauge lengths when the proposed method is used.For longer gauge lengths, the improvement is not noticeable.Comparing Figures 12(a) and 12(b), it seems that the errors are reduced at some gauge lengths when there is noise.Te reason is that the theoretical result was assumed as the true value for all wavelengths but it can only be used as the theoretical result at 1550 nm incident light wavelength, and the error defned is sensitive to the theoretical result that was chosen.Te noise used in the simulation may shorten the mean values from the mean simulated result to the theoretical result.
To show the change of the spatial frequencies, the period ratio which was defned as the ratio of the spatial frequency with the proposed method to the original spatial frequency is used in the red lines in Figures 12(c) and 12(d) correspondingly.Te blue lines in Figures 12(c) and 12(d) show the error ratios between the two methods.Te tendency of the error ratio increases when the gauge length increases and the decreasing tendency of the period ratio confrm the idea of increasing the accuracy of the strain detection by increasing the spatial frequency with noise level 0. Te period ratios are between 100% and 150%.Te small increase of the spatial frequency indicates that the new characteristics of the spectra generated by the proposed method only increase the high spatial frequency components slightly and the increased high spatial frequency components have a difculty to be distinguished from the noise if the noise level increases.As shown in Figure 12(d), the error ratios can be above 100% when there is 10% noise, which indicates that the proposed method is sensitive to noise but is benefcial to strain detection at short gauge lengths for low noise levels.
In summary, the characteristic spectral peaks within the measured spectral range are important for the spectral shift measurement under strain for NP-doped optical fbre sensing, especially for small gauge lengths because the backscattered light spectra show fewer characteristic spectral peaks for smaller gauge lengths compared with the cases for the larger gauge lengths as shown in Figure 7 (for the gauge length of about 0.1 cm gauge length cases, the spatial frequency is only about 1/nm − 1 which is 10 times lower than the cases for the gauge length of about 1 cm cases).With the newly proposed method, the accuracy of strain measurement will increase because of the increase in the characteristic spectral peaks.Terefore, the new method reduces the difculty of tracking the spectral shift for the small gauge lengths.However, it needs to be noticed that the new spectra will be more sensitive to the noise because the signal decreases and it may not be obviously benefcial for cases for Structural Control and Health Monitoring Figure 2(a) shows the structure of the tested optical fbre.Te incident light propagates through the optical fbre gauge length and is scattered by the randomly seeded NPs in the core.Te output signal is the accumulation of scattered light signals within the gauge length.Te procedure for calculating the strain values is shown in Figure 2(b).First, the NPs were seeded at random positions in the gauge.Te backscattered light spectra were calculated based on equation (

Figure 1 :
Figure 1: Te structure of the experimental setup.(a) Te structure of the distributed fbre optic sensing system.Te red dots show gold NPs randomly distributed in the core of the optical fbre.Te outer layers are the cladding and coating.(b) Te structure of the strain measurement system [37] with a specimen under load.

Figure 2 :
Figure 2: Te procedure of the simulations.(a) Te structure of the tested optical fbre.(b) Calculation procedure's blocks.

Figure 3 :
Figure 3: A case of the spectra of the random distribution of NPs in the optical fbre with a gauge length of 0.3207 cm for strain at − 1200 μϵ.(a) Te original spectrum and spectrum under strain.(b) Te results of the spectra with Fourier transform.Te black arrows show the spectral shift directions under − 1200 μϵ.(c) Te spectra modulated by the proposed method.(d) Te results of the modulated spectra with Fourier transform.Te black arrows show the spectral shift directions under − 1200 μϵ [45].

Figure 4 :
Figure 4: Te errors of the strains.Te cross-correlation results for the spectral range of (a) 10 nm, (b) 20 nm, and (c) 40 nm.Te numbers (− 1, − 2, − 3, and − 4) in fgures (a-c) correspond to the relative errors between the mean values and true values, the relative errors between the minimum values and the true values, the relative errors between the maximum values and the true values, and the relative errors between the mean values and true values with a threshold error of 0.5%, respectively [45].

Figure 9 :Figure 8 :Figure 10 :
Figure9: Te structure of the specimen on the four-point bending fatigue test machine.Te white dash hollow region is the capillary with 3 mm diameter.Te optical fbre sensor was installed in the middle of the capillary but it is not shown in this fgure.

Table 1 :
Light enhancement comparison with diferent methods.

Table 2 :
Te parameters used for the simulations.
Figure 7: Te relationship between the gauge length and the spatial frequency.Te spectral wavelength ranges are 10 nm, 20 nm, and 40 nm.Te red line shows the tendency [45].