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SECTION I

Optical frequency-domain reflectometry (OFDR) [1] has merits of simple configurations and high spatial resolution [2], [3]. The spatial resolution is inversely proportional to the wavelength tuning range of a tunable laser source (TLS) where the wavelength tuning nonlinearity during the fast tuning process is a major limitation for OFDR based sensing and the measurement is commonly not sampled with an equal frequency interval, which result in spatial resolution deterioration and make it difficult to detect the local spectrum shift and thus not capable to identify local temperature or strain variations.

Traditionally, the laser frequency tuning nonlinearity could be compensated by an auxiliary interferometer which provides an external clock to trigger data acquisitions based on the practically measured wavelength interval [4]– [5][6]. However, the maximum measurement length is limited by the optical path difference (OPD) between the two arms of the auxiliary interferometer that is four times the sensing fiber length according to the Nyquist sampling criteria [7]. Generally, OFDR with equal frequency-sampling method can achieve high spatial resolution (millimeters) for temperature and strain sensing with a total length of up to 70 m [18]. However, the traditional triggered interferometer approach cannot be implemented for long range temperature and strain sensing due to induced errors in the triggered signals from the auxiliary interferometer where polarization mode dispersion (PMD) and phase noise exist at a long distance for phase recovery of temperature and strain measurement [18].

Another method of suppressing the nonlinearity is to use laser sources of linear frequency tuning within a microwave frequency domain over a small tuning range by electrical optical modulator or compensate the tuning nonlinearity of the TLS frequency scanning [6]– [7] [8] [9] [10] [11] [12] [13] [14] [15][16]. One commonly used approach is to resample signals from OTDR with an equal frequency interval where the optical frequency can be extracted from the auxiliary interferometer [8], [9], [12], [13]. Although the spatial resolution could be improved, as the sensing range is independent on the OPD in the auxiliary interferometer, only short distances of tens of meters were experimentally demonstrated. Other approaches have been proposed by combining both improvement of the experimental setup and signal post-processing: High spatial resolution of 1 mm along a 2-km fiber has been presented by mixing a reference signal and an interference signal [6]. OFDR with high linear frequency tuning at equal time interval was achieved by using a single side-band modulator with a maximum sensing length of 40 km to identify the fiber ends [11]. A Deskew filter originally used in Radar systems was proposed and implemented with an 80 km range and 20 cm spatial resolution [13]. However, all the reported techniques merely focused on locating strong reflection points where no temperature or strain sensing results were demonstrated.

In this paper, we present OFDR-based distributed sensing with a measurement length over 300 m along a single-mode fiber using an interpolation algorithm, which allows a large wavelength tuning range to ensure the capability of a long sensing range with high temperature and strain resolution over high spatial resolution. The compensated tuning frequency is experimentally demonstrated to have comparable accuracy with the triggered interferometer OFDR at a short distance, and the sensing resolution deteriorates at a far distance. This work provides a new approach to achieve long range OFDR temperature and strain sensing that can be further improved by adopting a large buffer DAQ with powerful computer to calculate massive amount of the data for large optical frequency tuning range.

SECTION II

In order to compensate the nonlinear frequency tuning from tunable laser sources, an auxiliary interferometer is usually adopted to measure the nonlinearity in the frequency sweep of the TLS by obtaining phase information of an auxiliary beat signal through the Hilbert transformation. The auxiliary interferometer is basically a Mach–Zehnder interferometer (MZI). The beat signal generated from the auxiliary interferometer can be expressed as [12] TeX Source$$I (t) = U_{0}\cos \left(2\pi \tau v (t) \right)\eqno{\hbox{(1)}}$$ where $U_{0}$ is the amplitude of the beat signal, $\tau$ is the group delay difference between the two arms of the MZI, and $v(t)$ is the instantaneous optical frequency of the TLS which can be recovered by use of the Hilbert transformTeX Source$$v (t) = \tan^{-1}\left(I_{\rm H} (t)/I (t)\right)/(2\pi \tau)\eqno{\hbox{(2)}}$$ where TeX Source$$I_{\rm H} (t) = {\rm H}\left\{I (t) \right\} = U_{0}\sin \left(2\pi \tau v (t) \right).\eqno{\hbox{(3)}}$$

Since the rediscovered $v(t)$ is within the interval $(-1/(4\tau)$, $1/(4\tau))$, an unwrapping process is needed to remove the discontinuities in $v(t)$ and retrieve the true instantaneous optical frequency values which are sampled at equal time intervals between data points. Because of the virtual frequency tuning nonlinearity of the TLS that no linear relationship exists between $v(t)$ and $t$, signal fading occurs in the main interferometer that leads to deteriorated spatial resolution. Thus, an effective interpolation algorithm is needed to resample the beat signal obtained from the main interferometer with the same time interval. In the field of digital signal processing, many interpolation algorithms have been developed including linear interpolation, polynomial interpolation, spline interpolation, and so on [17]. Although linear interpolation fits curves by linear polynomials with high efficiency, its interpolation error is proportional to the square of the interval between data points, leading to high uncertainty in spatial resolution. High-order polynomial interpolation predicts data points with good accuracy; however, it only works for very few data points and thus is not favorable for processing large amounts of data points in OFDR. Cubic spline interpolation creates a smoother interpolated curve with a relatively small error and the interpolant is much easier to evaluate, thus providing an optimal cost-benefit tradeoff among available interpolation methods and is adopted in the current paper. Fig. 1 illustrates a method for smooth curve interpolation by means of cubic polynomial fitting with a set of continuous points to the raw data of uneven sample time interval: a smooth curve is interpolated where resampled data is in equal interval. It is noted that cubic spline interpolation yields quiet small interpolation error and predicts new data points with high accuracy, which satisfies the requirement of high spatial resolution Rayleigh backscatter profile used in distributed temperature and strain sensing.

The beat signal from OFDR is resampled to acquire an interference fringe with equal frequency spacing by using the cubic spline interpolation algorithm. A fast Fourier transform (FFT) is performed to obtain the power spectrum of reflected light at different scattering points by converting frequency-domain data into the time domain. In order to realize distributed measurement of temperature or strain, a reference trace is needed for relative phase change measurement between two different temperatures or strains at each location along the fiber under test (FUT). Segments of the Rayleigh backscatter pattern at the same positions are extracted from both the reference and the measurement profiles and then transformed back into the frequency domain using the inverse fast Fourier transform (iFFT). Through the iFFT data processing, a Rayleigh backscattering spectrum as a function of optical wavelength shift is obtained, Thus, the cross-correlation between the Rayleigh backscatter spectra of the reference and measurement segments presents a wavelength shift which is proportional to the changes in temperature or strain [7]. Distributed temperature and strain sensing can be achieved by calibrating these wavelength shifts.

SECTION III

Fig. 2 shows the experimental setup designed for distributed temperature and strain measurements over a long distance. The configuration of the OFDR system consists of a TLS (Newfocus TLB6600), a main interferometer with a polarization beam splitter (PBS) based polarization diversity receiver, an auxiliary interferometer with 100-m OPD to measure instantaneous optical frequency, photodetectors (PD) and a data acquisition card (DAQ, NI PCI-6115, maximum of 10 million samples per channel at 12 bit accuracy). One polarization controller (PC) within the main interferometer and another PC in front of the FUT are adopted to adjust light polarization states. An interference signal is from PBS to obtain a polarization-independent signal by calculating a vector sum of p- and s-components of the detected signals. The complex Fourier transform is carried out to convert the wavelength-domain signal into an optical time-domain reflectometry (OTDR) like trace along the FUT length. An Intel i7 3612qm mobile processor is employed for cubic spline interpolation and cross-correlation.

The FUT consists of three sections of different lengths of 68.0 m, 54.7 m, and 201.3 m concatenated by a pair of angled physical contact (FC-APC) connectors, a pair of physical contact (FC-PC) connectors, and another APC-APC connection, respectively. The fiber length before the first APC-APC connection is 7.4 m. A fiber knot with a small bending radius is deliberately induced at the end of the FUT to reduce the Fresnel reflection. The TLS sweeps from 1550 nm to 1556 nm. The DAQ card operates at an average sampling rate of 10 MS/s with a total 9.9 million sampling points. Different temperature and axial strain were applied to the FUT at different positions to calibrate the thermally and mechanically induced wavelength shift. The surrounding temperature was changed by using an oven at two different locations of 10.0 m extending over 1.8 m and 84.5 m over 40.0 m, while the temperature of the other part of the FUT is maintained at room temperature. The axial strain was changed by using translation stages to at two locations of 8.5 m and 75.5 m with the same length of 0.9 m, and the FUT was in a loose state of relative zero strain elsewhere.

SECTION IV

For long-range distributed sensing, good location accuracy is highly desired in addition to the high temperature and strain resolution detection of changes in environmental variables. Thus an OTDR-like trace is first acquired to locate temperature changing and strain applied positions on the optical fiber. The spatial resolution $\Delta Z$ is determined by the sweep frequency range $\Delta F$, following the formula of $\Delta Z = c/(2n\Delta F)$, where $c$ is the speed of light in vacuum and $n$ is the group index of the fiber [7]. The theoretical spatial resolution for 5.9 nm swept range is calculated to be 0.14 mm.

Fig. 3(a) and (b) show the Rayleigh backscatter signals as a function of distance along the FUT with and without using the cubic spline interpolation, respectively. It is obvious that the OTDR-like trace has been tremendously improved through the cubic spline interpolation to increase the effective spatial resolution. In Fig. 3(a), only one dominant Fresnel reflection peak is displayed at a rough location of the PC-PC connection and much positional information along the FUT is missing because of the relative low effective spatial resolution caused by the high nonlinearity of the TLS frequency scan. On the contrary, Fig. 3(b) distinctly exhibits three Fresnel reflection peaks induced by the first APC-APC, the PC-PC, and the second APC-APC connections at the positions of 7.4 m, 75.4 m, and 130.1 m, respectively, which are undetectable without using interpolation. Fig. 3(c)–(e) display true Fresnel reflection profiles of the above three connections, showing effective spatial resolution of 0.3 mm which is in excellent agreement with the calculated spatial resolution. Due to the high reflection as well as the PC reflector pair formed air cavity, some noise exists at the location of PC, therefore a larger distance window is adopted here to accommodate the complex signal occurring around the PC reflector, correspondingly leading to a larger division of the horizontal axis. The back reflection in the APC-APC connection is 17-dB less than that of the PC-PC connection, and the fiber knot at the fiber end is visible at 331 m with an 8-dB drop in the power, which validated a maximum $\sim$300 m sensing length. The spatial resolution can be determined by half maximum at full width (HMFW) at arbitrary APC reflection point. In Fig. 3(c) and (e), the spatial resolution is 0.14 mm at horizontal axe, where the HMFW consists of less than 2 sampling intervals ($\sim$0.28 mm). Therefore, a spatial resolution of less than 0.3 mm is achieved. In addition, several other peaks are also observed before the first APC-APC connection induced by the internal reflection points.

Fig. 4 shows experimental results of wavelength shift as the temperature was increased from 30 °C to 80 °C with an increment step of 5 °C. A total of 500 and 2300 data points at the positions of 10.0 m and 84.5 m along the OTDR-like traces are used to calculate the cross-correlation and calibrate the wavelength shift accordingly, equivalent to spatial resolution of 7.0 cm and 32.2 cm according to $\Delta {\rm Z}_{\rm sensing} = \Delta Z^{\ast}N$, where N is the number of data points. Fig. 4(a) presents wavelength shift of cross-correlation calculation with an increasing temperature along a section of fiber located at 10.0 m. Average wavelength shift is determined by averaging ten continuous points within the regions where the fiber were heated. Fig. 4(b) shows the average wavelength shift linearly changes with temperature. The thermal sensitivity coefficients at 10.0 and 84.5 m based on linear regression are 10.20 $\hbox{pm}/^{\circ}\hbox{C}$ and 10.02 $\hbox{pm}/^{\circ}\hbox{C}$, respectively. The errors at 10.0 and 84.5 m are calculated by taking the maximum error within the standard deviations from different temperatures to be 6.64 pm and 5.88 pm, which are converted to temperature accuracy of 0.65 °C and 0.59 °C, respectively.

Fig. 5 shows the experimental results of the strain induced wavelength shift of cross-correlation calculation. In Fig. 5(a), the wavelength shift under different strain, which was increased from 56.8 $\mu \varepsilon$ to 227.2 $\mu\varepsilon$ with an increment of 56.8 $\mu\varepsilon$, and from 227.2 $\mu\varepsilon$ to 568.0 $\mu \varepsilon$ with an increment of 113.6 $\mu\varepsilon$, is shown along the fiber at the position of 8.5 m with a length of 0.9 m. A total of 500 and 800 data points corresponding to strain spatial resolution of 7 cm and 11.2 cm are used to determine wavelength shifts at 8.5 m and 75.5 m. Fig. 5(b) presents the average wavelength shift with respect to varying strain at 8.5 m and 75.5 m where linear regression is applied and the strain sensitivity coefficients are calculated to be 1.20 $\hbox{pm}/\mu\varepsilon$ and 1.22 $\hbox{pm}/\mu\varepsilon$, respectively. The wavelength shift measurement errors at 8.5 m and 75.5 m are 1.82 pm and 2.80 pm, corresponding to strain accuracy of 1.5 $\mu\varepsilon$ and 2.3 $\mu \varepsilon$, respectively.

In temperature and strain sensing experiments, it is found that the sensing resolution deteriorates as the fiber length extends. In order to determine the actual sensing resolution, the number of data points needed for obtaining an optimum cross-correlation figure with a minimum side peak suppression ratio of 2 is calculated. The cross-correlation figures at three different locations of 10 m, 140 m, and 310 m using 500, 2800, and 6000 data points in the temperature measurement are illustrated in Fig. 6, which indicates that the predicted temperature spatial resolution are about 7 cm, 39 cm, and 84 cm, respectively, as shown in Figs. 6(a)–(c). In addition, the temperature spatial resolution at different locations (10, 50, and 150 m) along the FUT as a function of the wavelength tuning speed is shown in Fig. 7. It is found that at the same locations a higher wavelength sweeping speed (e.g., 10 nm/s) requires more data points for cross-correlation compared to a lower wavelength sweeping speed (e.g., 6 or 8 nm/s), indicating reduced spatial resolution with increasing wavelength tuning speed.

This spatial resolution deterioration effect can be explained in the following three aspects: First, although the cubic spline interpolation can predict data points with relative high accuracy compared to other well-known algorithms, the prediction errors still exist in the resampled beat signal and accumulate at the far end of the FUT, leading to the deterioration in spatial resolution. Secondly, unlike the long range OFDR locating applications, temperature and strain sensing requires more accurate local Rayleigh backscattering profile to achieve precise wavelength shift value. However, the increased phase noise at a longer distance may disturb the local spectrum and more data points are needed for cross-correlation to obtain a good signal of high SNR. Lastly, the linewidth of the TLS operating at the fast wavelength tuning speed of 5.9 nm/s (737 GHz/s frequency sweeping speed, equivalently) increases relative to the case of a low sweeping speed due to the increasing phase noise during the lasing process, which leads to a reduced coherent length [19]. Therefore, one must carefully select an appropriate tuning speed in order to maintain the accuracy of Rayleigh backscattering profile. Although a smaller tuning speed can reduce the phase noise, the spatial resolution will correspondingly decrease given the fixed sampling points of the DAQ. Considering above factors, a $\sim$300 m SMF is deliberately chosen for demonstration in this work.

SECTION V

In conclusion, distributed temperature and strain sensing are realized based on an OFDR system with a measurement length of more than 300 m by using a cubic spline interpolation algorithm which effectively compensates the frequency tuning nonlinearity of the TLS. The temperature and strain sensitivities are 10 $\hbox{pm}/^{\circ}\hbox{C}$ and 1.2 $\hbox{pm}/\mu\varepsilon$, respectively, with measurement accuracies of 0.7 °C and 2.3 $\mu\varepsilon$ at spatial resolution of 7 cm. The sensing resolution deteriorates at far distances due to increased phase noise from the laser tuning process. To further improve the sensing distance and spatial resolution, one may use a TLS with a large tuning range and a slower tuning speed, as well as a DAQ card of a large on-board memory buffer combined with a powerful computer. The proposed technique enables an extended sensing distance with excellent spatial resolution, spectral sensitivity, and measurement accuracy over other traditional nonlinearly-triggered OFDR systems, and has great potential in a variety of practical sensing applications such as structural health monitoring of civil infrastructure systems.

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