1. Introduction

Optical frequency domain reflectometry (OFDR) is a coherent technique for measuring the spatial distribution of backscattering in optical fiber [1]. Originally used to locate losses, it was soon realized that, because it measured the phase as well as the amplitude of the backscatter, OFDR could be used as an interrogation technique for distributed optical fiber sensing [2]. In distributed fiber sensing, the optical fiber is used as both the transmission line and the transducer, and the measurand can be measured at each point along the fiber. For recent reviews of distributed optical fiber sensing, see Refs. [3] and [4].

One major driver of research into and development of distributed optical fiber sensing is the ability to perform structural health monitoring (SHM) of modern civil and transport structures [5,6]. Condition-based SHM promises to increase operational availability of modern structures while decreasing the time and dollar costs of planned and unplanned maintenance. The mechanical strain and thermal stress states of the key structural components must be measured continuously for both direct fatigue detection and to inform digital twin models. Due to the complex nature of modern structures, dense sensor placement ($\sim$mm) over a moderate sensing range (tens or hundreds of meters) is often necessary. Further, common SHM applications, such as those on transport vehicles [7,8] or machine parts [9], require sensors that are reliable in harsh environments. Conventional strain sensors that consist of a few electronic (piezoelectric or resistive) sensors placed sparsely around the structure cannot meet these demands.

Distributed fiber optic sensing provides a means to measure the strain and temperature profile of complex structures, enabling continuous and permanent condition and structural health monitoring. Fiber sensors are lightweight and durable, and many sensors can be multiplexed on a single optical fiber. For SHM applications, OFDR is often the sensing modality of choice because of its high spatial resolution ($\sim$mm), strain resolution ($\sim \mu \epsilon$), and moderate sensing range (tens to hundreds of meters). The design of OFDR systems is well-understood; OFDR-based distributed strain (and temperature) sensing systems are commercially available. Nevertheless, one important decision when using a distributed strain sensing system is the choice of the sensing fiber.

Single Mode Fiber (SMF) is often used for distributed strain sensing because it is inexpensive and readily available [10]. However, the signal-to-noise ratio (SNR) for a strain measurement via Rayleigh back scatter is low because the signal is weak. Additionally, OFDR measurements of Rayleigh back scatter are relatively slow because a wide wavelength sweep is required. Further, the dynamic range is limited by the decrease of overlap between the reference spectrum and measurement spectrum at high strains [3,11].

All-grating fiber (AGF) is an alternative sensing fiber with stronger back scattering. All-grating fiber consists of many weak, nearly-identical fiber Bragg gratings spatially multiplexed down the length of the fiber, typically produced on the drawtower [12–14]. The AGF used here is commercially available AGF from FBGS (AGF-LBL-1550-80).

In this paper we study the limits of performance of OFDR systems using all-grating fiber. We develop models to quantify how both the Born approximation and the oft-neglected "auto-correlation term" limit the sensing range of AGF when measured with OFDR. The Born approximation breaks down when the gratings are long and/or strong enough so that multiple reflections within the grating are non-negligible. The auto-correlation term is a non-interference term that is inherent to OFDR measurements. It can become significant when measuring a long, coherent structure such as AGF with OFDR. Where possible, we compare our models with OFDR measurements. Our models consider the critical fact that the Bragg gratings are not identical, but have a distribution of center wavelengths. We find that while the auto-correlation term significantly impacts OFDR measurements of the complex coupling coefficient, it does not cause a significant error in distributed strain or temperature measurements. On the other hand, the breakdown of the Born approximation limits the sensing range of AGF.

We conclude by proposing two novel solutions, one of which utilizes commercially available fiber, to extend the sensing range of AGF by controlling the properties of the non-identical Bragg gratings.

2. Distributed strain sensing with optical frequency domain reflectometry

The formalism of fiber optic distributed strain sensing with OFDR has been developed in terms of the coupled mode theory of fiber Bragg gratings (FBGs) [2], in which the properties of an FBG can be completely described in terms of a complex coupling coefficient $q(z)$. This formalism has been extended to Rayleigh back scattering in single mode fiber by realizing that the defects causing the Rayleigh back scatter are frozen into the fiber. One can then consider the defects as an FBG with random complex coupling coefficient [2].

The complex coupling coefficient $q(z)$ is related to the physical properties of the grating: the amplitude of $q(z)$ is related to the strength of the modulation of the index of refraction of the optical fiber, and the derivative of the phase of $q(z)$ is determined by the effective grating period [15]. When the optical fiber is strained at the location of the FBG, the effective grating period changes, causing a change in the phase of $q(z)$. Therefore, measuring the strain as a function of position along the fiber is reduced to measuring $q(z)$.

When the scatterers (e.g., FBG or Rayleigh backscatter) are weak enough that multiple reflections can be neglected, the measurement of the $q(z)$ is reduced to a measurement of the (complex) impulse response $h(z)$ of the FBG [15]:

Neglecting multiple reflections is referred to as the Born approximation. The Born approximation breaks down for strong and/or long gratings (i.e., when $|q|L \not \ll 1$, where $L$ is the grating length). In these cases, computationally expensive inverse scattering techniques such as discrete layer peeling [16] or Gel’fand-Levitan-Marchenko-based algorithms [17] must be used to determine $q(z)$ from $h(z)$.

There are several different techniques for measuring $h(z)$ of an optical fiber. OFDR is often used when high spatial resolution (as small as 10 $\mu m$) and short to medium lengths (up to a few hundred meters) are required. OFDR is a coherent frequency-domain technique; the impulse response of an optical fiber is determined by measuring the complex frequency response and exploiting the Fourier transform relationship between the impulse response and the frequency response [1].

In OFDR, a fiber-coupled CW tunable laser source (TLS) performs a linear frequency sweep in the 1550 nm range. The TLS output is the input to an interferometer, as shown in Fig. 1(a). Here, we depict an unbalanced Michelson interferometer for convenience, though a Mach-Zehnder configuration can also be used. In either case, the analysis proceeds similarly.

 

Fig. 1. a) Schematic of OFDR system using an unbalanced Michelson interferometer. Mach-Zehnder configurations are also often used. (TLS -- tunable laser source). The distance down the fiber from the coupler to the front of the all-grating fiber is denoted $z_{FBG}$, and the distance from the coupler to the reference mirror is $z_{Ref}$. b) Simulated and measured smoothed impulse response of 20 m of all-grating fiber. In Fig.~1b, the position $z=ct/2n_{g}$. The measurement was performed using OFDR (Luna OBR). The simulated impulse response agrees remarkably well with the measurement in the autocorrelation, grating, and ghost grating regions. The disagreement in the region between 20m and 40m occurs because of the Rayleigh back scatter in the measurement, which we do not model. Inset) Unsmoothed measurement (shown in orange) of the impulse response of the first five gratings.

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Consider a laser field at a fixed frequency $E_{laser}=E_0\exp [i\omega t]$. At the coupler the electric field reflected by the reference mirror is given by $E_{Ref}=\frac {1}{2}aE_0\exp [i\omega (t-\tau _{Ref})]$, where $a$ is the reflectivity of the mirror. The round-trip light propagation time from the mirror to the coupler is given by $\tau _{Ref}=2n_{g}z_{Ref}/c$, where $n_g$ is the group index and $z_{Ref}$ is the distance down the fiber from the coupler to the reference mirror. The light reflected by the measurement arm is given by $\frac {1}{2}aE_0r(\omega )\exp [i\omega (t-\tau _{FBG})],$ where $r(\omega )$ is the complex reflectivity of the entire FBG array. The round-trip light propagation time from the front of the FBG array to the coupler is given by $\tau _{Ref}=2n_{g}z_{FBG}/c$, where $z_{FBG}$ is the distance down the fiber between the coupler and the front of the FBG array.

In OFDR, the wavelength of the tunable laser source is swept linearly in time, so that the intensity of the light at the photoreceiver as a function of optical frequency $\omega$ is measured:

In order to obtain the impulse response (which is related to $q(z)$ by Eq. (1)), we take the inverse Fourier transform of the measured intensity in post-processing. The inverse Fourier transform of Eq. (2) is given by

In Eq. (3) the first term contributes only for $\tau =0$. The second term is the autocorrelation of the impulse response and is typically neglected on the assumption that it contributes only for small $\tau$. Because this term is important for our work, we explicitly define the autocorrelation term:

The fourth term is eliminated when $\tau >-\Delta \tau$ because the argument corresponds to a position outside of the fiber. Therefore, by considering only $\tau > \Delta \tau$ and making the transformation $z=ct/2n_{g}$, we obtain the impulse response $h(z)$ from the inverse Fourier transform of the OFDR measurement. Then the Born approximation or other inverse scattering method can be used to obtain $q(z)$ (and therefore the strain) from $h(z)$.

A measured impulse response with the contributions from different terms is shown in Fig. 1(b).All measurements in this work, including this one, were taken with the Luna OBR 4600 with the following settings: wavelength scan range: 10 nm, center wavelength: 1550 nm, gain: 0, measurement range: 70m.

From the preceding discussion, it is clear that there are two standard approximations in computing the complex coupling coefficient $q(z)$ from the OFDR measurement of $I_d(\omega )$: A. $A(\tau )\rightarrow 0$ (which we refer to as the autocorrelation approximation) and B. the Born approximation (Eq. (1)).

In the next section, we examine the validity of these two approximations for all-grating fiber.

2.1 OFDR measurements of all-grating fiber

All-grating fiber (AGF) consists of many weak nearly-identical FBGs spatially multiplexed down the length of the fiber. AGF provides an increased back scatter signal and allows for rapid data collection because the wavelength sweep range can be much smaller than for Rayleigh scattering [3,11]. If $N$ is the ratio between the wavelength scan range required for an accurate Rayleigh scatter measurement and the wavelength scan range required for an accurate AGF measurement, the rate at which distributed measurements on AGF can be performed is greater by a factor of order $N$ compared to Rayleigh scatter. Additionally, the data processing can be also performed faster by a factor of order $N\log (N)$, since all major processing steps (inverse Fourier transform to obtain the impulse response, Fourier transforms to obtain the localized reflectivity, and the cross-correlation to obtain the localized wavelength shift) rely on the fast Fourier transform algorithm. Further, the tools available from FBG analysis allow for improved spatial resolution and processing efficiency [19]. Here we consider commercially available AGF from FBGS (AGF-LBL-1550-80); each FBG is 9 mm long with reflectances less than $0.1\%$ and 1 mm spacing between FBGs.

An OFDR measurement of the impulse response of 20 m of AGF with 40 m of lead-in SMF is shown in Fig. 1(b). A few features are noteworthy. The first is the signal that corresponds to reflections from the AGF, between 41 m and 61 m. This is the part of the impulse response from which we want to determine $q$. The second is the well-known “ghost-grating” tail starting near 60 m. This signal appears to come from a position beyond the end of the AGF, where $|q|$ should vanish and there should be no reflections. This discrepancy is due to multiple reflections within the AGF, in violation of the Born approximation. In the typical situation in which only triple-reflections are significant, the “noise” level due to the “ghost grating” tail gives the approximate error in the measurement of $q$ due to multiple reflections [20]. In principle, the error due to the ghost gratings can be removed by inverse scattering techniques; in practice the high computational complexity of these techniques has led to the validity of the Born approximation typically being considered the limiting factor in the length of AGF that can be used for strain sensing with OFDR.

The third noteworthy feature occurs over the first 20 m of the impulse response in Fig. 1(b) and is due to the autocorrelation term in Eq. (3). This term has received very little attention in the literature, but, as we will show, can limit the accuracy of $q(z)$ measurements. Therefore, it is essential to understand the limitations on OFDR strain measurements imposed by this term and how to mitigate them.

We emphasize the difference between the autocorrelation term and the Born approximation. The autocorrelation term is simply the inverse Fourier transform of the power spectrum of the sensing arm of the interferometer. This term is due only to the light that is reflected by the sensing arm of the interferometer. It is inherent to OFDR measurements without balanced detection; that is, it is inherent to the measurement apparatus. As the inverse Fourier transform of the reflectivity spectrum, a grating with strong reflection over a narrow bandwidth would result in a broad auto-correlation term. In most analyses of OFDR, the autocorrelation term is typically neglected on the assumption that it contributes only for small distances down the fiber. However, as we demonstrated in Fig. 1(b), for AGF the autocorrelation term does not decay rapidly and can cause significant error in OFDR measurements. The autocorrelation term always begins at $z=0$, and can therefore be eliminated by increasing the length of the lead-in fiber (at the cost of a decreased sensing range for a fixed $\omega$ resolution). Alternatively, as stated above, the autocorrelation term can be eliminated by using an interferometer with balanced detection.

The Born approximation, on the other hand, is the assumption that there are no multiple reflections in the grating, or, equivalently, that $q(z)\propto h^*(z)$. Like the autocorrelation approximation, this assumption breaks down for long and/or strong gratings. Unlike the autocorrelation approximation, the accuracy of the Born approximation is not affected by the length of the lead-in fiber or the presence of balanced detection. Instead, in cases where the Born approximation breaks down, $q(z)$ can be retrieved from $h(z)$ using inverse scattering methods.

2.2 AGF model

We develop a model of $q(z)$ for AGF. This allows us to separate out the effects of the autocorrelation and Born approximations and to exclude the noise sources that are present in experimental measurements. It also allows to determine the effect of varying grating parameters without having to write the gratings.

From $q(z)$, the complex reflectivity $r(\omega )$ can be computed using any of a number of scattering techniques such as the transfer matrix method or inverse discrete layer peeling [15]. Here, we use inverse discrete layer peeling (iDLP). The signal that would be measured at the detector can then be obtained by Eq. (2); the impulse response that would be measured ($h_{meas}$) is then obtained from the inverse Fourier transform of the detector measurement. The autocorrelation term can be separated out in simulation by retaining only the second term of Eq. (2). Similarly, the true impulse response can be obtained by performing the inverse fast Fourier transform on $r(\omega )$.

We model the AGF as a series of individual gratings of equal strength (measurements show that $|q|\approx 3$m$^{-1}$) and length ($L=9$mm). The gratings are spaced 1mm apart. While previous works [21,22] model the gratings as identical in wavelength, we find that this leads to a severe overestimate of the effect of multiple reflections. Therefore, we randomly select the center wavelength of each grating from a Gaussian distribution with mean 1550nm and standard deviation 55pm; these parameters were measured from our AGF as follows: We measured several different short (roughly 1m) sections of AGF using the Luna OBR 4600. We divided up the OBR measurement into time domain segments such that each segment represented one FBG from our AGF. We performed a Fourier transform on each segment in order to obtain its reflection spectrum; these spectra were then used to determine the distribution of wavelengths. The grating center wavelength determines the derivative of the phase of $q(z)$ [15]:

The impulse response obtained from our model of 20m of AGF is shown in Fig. 1(b). Our model agrees excellently with the complex magnitude of the OFDR measurement in all three key features: the autocorrelation signal, the gratings themselves, and the ghost gratings (Born approximation error). With our model validated, we can now use it to examine the autocorrelation approximation and the Born approximation in detail.

2.3 Effect of approximations on measurements of $q(z)$

OFDR is often used to determine the complex coupling coefficient $q(z)$ from measurements of the grating reflectivity; this procedure is often called grating synthesis or reconstruction [16,17,23]. In this section, we study the impact of the autocorrelation and Born approximations on the accuracy of the reconstruction of the grating $q(z)$ in OFDR measurements.

First, we consider the autocorrelation approximation. The autocorrelation term is not an interference term and can therefore be eliminated by using an interferometer with balanced detection, as suggested in Ref. [15]. However, in practice (including in commercially available interrogators from LUNA and Sensuron) balanced detection is rarely done due to the additional costs. An inexpensive way to deal with the autocorrelation term is to add sufficient lead-in SMF so that the autocorrelation term is negligible in the sensing region, as was done for the impulse response shown in Fig. 1(b). However, this can limit the sensing range.

The autocorrelation term itself is independent of the location of the grating in the fiber: it always peaks at $z=0$ and decays as $z$ increases. Therefore, the effect of the autocorrelation term on the measurement of the grating $q(z)$ depends on the position of the start of the grating in the fiber (relative to the reference mirror) $\Delta z = z_{FBG}-z_{Ref}$; as $\Delta z$ increases, the error due to the autocorrelation term decreases because the overlap between the autocorrelation term and the grating impulse response decreases.

We consider 10m of AGF with $\Delta z=1$m, with the other parameters the same as described in Section 2.2. We separate out the effect of the autocorrelation term from all other approximations and noise sources by adding the autocorrelation of the computed impulse response $A(z)$ directly to the known $q(z)$ of the grating. The complex magnitude $|h(z)+A(z)|$ is shown in Fig. 2(a). In Fig. 2(b) the error in the measurement of the maximally reflected wavelength (computed from the phase of $q(z)$ by Eq. (5)) is shown. Also shown are the moving root-mean-square (RMS) error, computed over a moving window of 0.5m. There is significant error due to the autocorrelation term for small $z$, and this error decreases as $z$ increases. Figure 2(c) shows the impact of the autocorrelation on $q(z)$ near $z=0$, where the error is greatest. Significant error is introduced into the measurement of both $|q|$ and the effective Bragg wavelength (which is related to the phase of $q(z)$). The bottom panel shows $\overline{\Delta \lambda }$, the effective Bragg wavelength averaged over a single grating. Figure 2(d) shows the impact of the autocorrelation on $q(z)$ near the end of the 10m AGF, where the impact is significantly reduced by the decay of the autocorrelation. There is some error in the effective grating wavelength (middle panel), but this is reduced by averaging over each grating (bottom panel).

 

Fig. 2. Effect of the autocorrelation approximation on simulated measurements of $q(z)$ for 10m of AGF when $\Delta z=1$m. a) Simulated $|h(z)+A(z)|$ for 10m of AGF when $\Delta z=1$m (including the autocorrelation term and ghost gratings). b) Error and local RMS error in the determination of the local effective Bragg wavelength due to the autocorrelation term. The local RMS error is calculated with a moving window of width 0.5m. c) Top: Comparison of $|q(z)+A(z)|$ with the true $q(z)$ at the beginning of the AGF. Middle: Error in the grating phase measurement due to the autocorrelation term, scaled to represent error in the wavelength measurement. Bottom: $\overline{\Delta \lambda }$, the error in the effective Bragg wavelength averaged over a single grating. The autocorrelation term causes severe degradation of the determination of $q(z)$ at the start of the grating. d) The degradation of the determination of $q(z)$ due to the autocorrlation term is significantly reduced after 10m, especially when the phase derivative is averaged over an entire grating. In the legends, $|\cdot |$ indicates the complex magnitude.

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For AGF with these parameters, accurate $q(z)$ measurements can be obtained with $\Delta z\geq 10$m for 10m of AGF. We have simulated lengths up to 50m of this AGF, and we conclude that the amount of lead-in SMF required is approximately equal to the length of AGF to be measured up to about 30m of AGF. For longer gratings, the length of the lead-in fiber must increase even further. Thus, the autocorrelation term leads to a sensing range reduction by a factor of two when using AGF as the sensing fiber for fiber lengths under 30m.

Next, we consider the effect of the Born approximation on measurements of $q(z)$. In order to do this, we use the complex reflectivity (computed from $q(z)$ by iDLP) and Eq. (2) to obtain the optical intensity at the detector. An inverse fast Fourier transform is then used to obtain $h_{meas}$, which contains the effects of the autocorrelation term and the Born approximation. For this simulation, we use $\Delta z=10$m so that the autocorrelation term has little effect on the grating measurement for our 10m grating. This is verified by looking at the complex magnitude of the impulse response $h_{meas}$, as shown in Fig. 3(a). Figure 3(b) shows how the error in the phase derivative of $q(z)$ (converted to a wavelength by Eq. (5) and averaged over each grating) increases with the distance from the start of the grating. A more detailed look at the measurement at the start of the grating is shown in Fig. 3(c). It is clear that the measurement of the magnitude and phase of $q(z)$ are accurate at the start of the grating. On the other hand, the Born approximation causes significant error in the measurement near the end of the 10m grating, as shown in Fig. 3(d). The error in the phase derivative (again represented as a wavelength) is particularly large (middle panel), although this error is reduced when averaging over each grating (bottom panel).

 

Fig. 3. Effect of the Born approximation on simulated measurements of $q(z)$ for 10m of AGF when $\Delta z=10$m. The autocorrelation term has little effect because $\Delta z=10$m. a) Simulated $h_{meas}$ for 10m of AGF when $\Delta z=10$m (including the autocorrelation term and ghost gratings). b) Error in the determination of the local effective Bragg wavelength due mainly to the Born approximation. The local RMS error is calculated with a moving window of width 0.5m. c) Top: Comparison of $h_{meas}$ with the true $q(z)$ at the start of the grating. Middle: Error in the grating phase measurement, scaled to represent error in the wavelength measurement. Bottom: $\overline{\Delta \lambda }$, the error in the effective Bragg wavelength averaged over a single grating. The autocorrelation term causes severe degradation of the determination of $q(z)$ at the start of the grating. The Born approximation causes little degradation of the determination of $q(z)$ near $z=0$ because multiple reflections do not play a role near the beginning of the grating. d) The degradation of the determination of $q(z)$ due to the breakdown of the Born approximation is significantly worse near the end of the grating. In the legends, $|\cdot |$ indicates the complex magnitude.

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In the absence of the autocorrelation term, more accurate inverse scattering techniques such as discrete layer peeling [16] can be used in place of the Born approximation to remove the effects of multiple reflections. However, when the autocorrelation term is present in the measurement, these inverse scattering techniques can fail because errors build up from the front of the grating to the back of the grating. Our simulations (which have no noise or polarization effects) find that discrete layer peeling gives inaccurate grating reconstruction for 20m of AGF when $\Delta z<10$m. The noise and polarization rotation that are inevitably present in measurements will decrease this threshold for failed grating reconstruction.

2.4 Effect of the Born approximation on distributed temperature measurements

In the previous section, we discussed the effect of the autocorrelation and Born approximations on the measurement of $q(z)$. In this section, we study the effect of these approximations on distributed temperature measurements. The discussion and results apply equally to distributed strain measurements, since strain affects the fiber in the same way.

The measurement of $q(z)$ is an absolute measurement, in the sense that a measurement of the $q(z)$ of a single fiber characterizes the gratings in that fiber. A temperature or strain measurement, on the other hand, is a relative measurement: the temperature or strain is determined from the change in the phase of $q(z)$ relative to a reference measurement. Thus, it may be expected that these approximations could affect temperature or strain measurements differently than they affect measurements of $q(z)$.

In distributed temperature sensing, the temperature profile as a function of the position along the grating is measured. Different temperature profiles may affect the autocorrelation term and the location and strength of multiple reflections in different ways. In the worst case scenario, one could measure a temperature profile that forces all the gratings to have the exact same center wavelength, maximizing both the autocorrelation term and the effect of multiple reflections. However, such a temperature profile seems unlikely in a practical setting. Here, we consider a constant temperature change of $14^{\circ }$ C applied over a 10m length of fiber beginning at $\Delta z=40$m relative to the reference reflector, and the temperature change is zero elsewhere. This choice was made to minimize the influence of the autocorrelation term.

We simulate a change in temperature by

For AGF, the temperature shift in each grating can be computed directly from the phase derivative of $q(z)$ [19]; this technique has the advantage of high spatial resolution and computational efficiency. However, the phase derivative technique limits the strain dynamic range and is sensitive to noise and the polarization effects in OFDR measurements. Therefore, in this work, we compute the temperature shift from the impulse response using a standard window-then-FFT procedure [24], which we briefly summarize. The impulse response is divided up into 2cm segments, and the reflection spectrum of each segment is obtained by Fourier transform. The spectral shift due to the temperature change is determined by performing a cross-correlation on corresponding segments from the reference and measurement impulse responses. The temperature shift of each segment can be computed by $\Delta T = K_T\Delta \lambda /\lambda$. Throughout the rest of this work, we will refer to this procedure as the "window-FFT-xcorr" method.

The spectral shifts for our simulated AGF are shown in Fig. 4(b). The error increases with distance down the fiber. No autocorrelation term, polarization effects, or noise sources are included in our simulations, so we can attribute the error to the breakdown of the Born approximation.

 

Fig. 4. Effect of the Born approximation on temperature measurements of AGF when $\Delta z=40$m. The autocorrelation term has little effect because $\Delta z$=40m is significantly larger than the grating length. a) OFDR measurement of 20m of AGF, where temperature change of $14^{\circ }$ C (relative to the reference measurement) corresponding to a 200pm shift has been applied to the first 9m of AGF. The -25pm spike around 13.9m may be due to a small strain induced on the fiber during the measurement. b) Results of a simulated OFDR measurement of AGF using our model. A 200pm shift was applied to the first 10m of the 30m AGF using Eq. (5). c) Zoom in of Fig. 4(a). The RMS error is around 5pm. d) Zoom in of Fig. 4(b). The RMS error is at about 5pm from 10m-20m, then increases significantly once large error spikes begin to occur after 20m.

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We perform a distributed temperature measurement to validate our simulations. A 40m SMF patchcord was spliced to 20m of AGF; therefore, the autocorrelation term is insignificant because $\Delta z$=40m is significantly longer than the AGF. The first 9m of AGF was spooled and placed in a pool of water cooled to 4.5$^{\circ }$ C; the rest of the AGF was kept on a spool at 18.5$^{\circ }$ C. A reference OFDR measurement was taken with the LUNA OBR using a 10nm wavelength sweep. Then the water was warmed to 18.5$^{\circ }$ C, and a second OFDR measurement was taken. The results of the spectral shift computation on the OFDR measurements are shown in Fig. 4(a). The measurement is reasonably accurate over the full 20m, with the RMS error increasing from about 2pm to 4pm near the end of the grating. The simulation also shows a qualitatively similar increase in RMS error as the distance down the fiber increases. Additionally, after 20m large error spikes appear in the simulation that are due to the further breakdown of the Born approximation.

In principle these errors can be reduced by inverse scattering techniques; however, the difficulties presented by noise, polarization rotation and the computational expense of the techniques render the Born approximation essential for distributed sensing. In the following sections, we discuss other ways to decrease the noise in OFDR distributed strain and temperature measurements.

We note that we have performed measurements and simulations for smaller values of $\Delta z$. When the “window-FFT-xcorr” method is used to determine the strain or temperature along the fiber, we find minimal error (at most 5pm) due to the auto-correlation term. We believe the error due to the auto-correlation term is minimal because of the relative nature of the measurement: The auto-correlation does not change much if strain is applied to the fiber, and is “filtered out” by the relative nature of the measurement. Therefore, even though the measurement of $q(z)$ is significantly impacted by the auto-correlation term, there is not a significant amount of error induced in the strain or temperature measurement.

3. Discussion

Our results reveal the tradeoffs that must be weighed when choosing which sensing fiber to use with OFDR. AGF permits the best spatial and strain resolution, the fastest update rate, and the highest SNR, but is limited by the Born approximation to about 10m in range. Further, even in a situation where inverse scattering is computationally feasible and the range can be extended, a large lead-in fiber is required to reduce the error from the autocorrelation approximation. This increases the required ADC sampling rate and data throughput. On the other hand, SMF permits a longer sensing range because the autocorrelation and Born approximations are insignificant. However, the maximal interrogation rate is reduced because a wide wavelength sweep is required. Further, the spatial resolution is decreased and the data processing is more computationally expensive compared with AGF.

A possible middle ground would be some type of enhanced backscatter fiber. This fiber would have stronger backreflection than SMF but the reflectors would be incoherent. There are many ways to enhance backscattering. Reduced-cladding single mode fiber can provide up to a fourfold improvement in strain sensitivity over regular SMF [25]. Doping the fiber core with gold nanoparticles has been shown to provide a 30dB enhancement of backscatter amplitude at the cost of increased loss (2.8 dB per m), which severely limits the sensing range [26]. Weak random gratings (created by randomly dithering the phase during writing) have also been shown to increase the Rayleigh backscatter by up to 50dB over a wide bandwidth with loss of as low as 0.15dB/m [27]. Broadband enhanced scattering fiber made from overlapped chirped gratings has shown promise as a specialty distributed sensing fiber [28,29]. This fiber has similar backscatter strength ( 20 dB over Rayleigh) as the AGF used here, but has a less coherent grating structure that may reduce the strength of the autocorrelation term and also lead to a reduction in the multiple scattering that causes Born approximation error, leading to an increased sensing range.

Our work has revealed some ways to improve the distributed strain and temperature sensing capabilities of AGF fiber. The range of AGF is limited mainly by the presence of multiple reflections. Multiple reflections can be reduced by making weaker gratings, but this reduces the SNR and creating weaker gratings on the draw tower is technologically challenging.

Another way to mitigate the noise from multiple reflections is to increase the width of the center wavelength distribution of the gratings. A combination of slightly weaker gratings and a wider spread in the wavelength distribution may have been used in Ref. [28] to obtain a reduction in multiple reflection noise. In Sections 2.3 and 2.4, we studied AGF with gratings with center wavelengths chosen from a normal distribution centered around 1550nm with standard deviation $\sigma =0.055$nm. Here, we study the effect of changing $\sigma$ on the measurement range of AGF.

We now use a previously developed model [30] to estimate the maximum optical power that appears to come from a distance $z$ down the fiber, but is actually due to triple reflections. We use this optical power $P_{max}$ as an approximation for error in the distributed strain/temperature measurement due to the Born approximation. Later, we will compare with our own model, which considers reflections of all orders and provides the error in the distributed strain/temperature measurement directly.

The maximum reflected optical power from triple reflections that appears to come from the $k^{th}$ grating can be computed as [30]

We make the approximation that each individual grating is weak and has a constant $|q|$ so that

For our model, the grating width $\ell = 9$mm, and the grating strength $|q|=3$m$^{-1}$, as before. We draw the $\lambda _k$ from a normal distribution centered at 1550nm with standard deviation $\sigma$. The result of computing Eq. (7) for a 10m grating for different $\sigma$ is shown as solid lines in Fig. 5(a). For $z< 10$m, $P_{max}^{(3)}$ grows quadratically with $z$, as expected, since the number triple reflections also grows quadratically with $z$: Eq. (7) shows that the measurement of the $k^{th}$ grating is affected by $(k-1)(k-2)/2$ triple reflections. For $z> 10$m (i.e., beyond the end of the grating region), the triple reflections appear to be coming from beyond the grating region and therefore do not cause any strain error; the max power from these reflections decays quadratically.

 

Fig. 5. Effect of the width of the wavelength distribution on multiple reflections. a) Maximum reflectance from each position on 10m of AGF. Solid lines show direct simulation results of Eq. (7), and dashed line shows the best-fit quadratic. From top curve to bottom, $\sigma$ increases from 0pm to 100pm. The inflection point occurs at the location of the end of the grating, and the decrease is quadratic as well. (It appears sudden because it is plotted on a log-log scale.) b) Simulation of strain measurement with AGF with $\sigma =100$pm. c) Zoom in of Fig. 5(b). The RMS error is less than 2.5pm.

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It is clear that increasing $\sigma$ decreases the noise from triple reflections for a given $z$. The dashed lines in Fig. 5(a) show the best quadratic fit for each $\sigma$. An extrapolation of these fit lines predicts that a 20m AGF with $\sigma =0.1$nm would have the same amount of noise from triple reflections as a 10m AGF with $\sigma =0.055$nm, as is the case for the AGF from FBGS.

We verify this by direct simulations, shown in Fig. 5(b). A $q(z)$ representing AGF with $\sigma =100$pm was created, and its impulse response was computed using iDLP. Then a constant 160 microstrain was applied to the $q(z)$ over a 10m region using Eq. (6), and the impulse response was again computed. The measured strain was computed from the impulse responses using the “window-FFT-xcorr” method described in Section 2.4. It is clear that the measurement error from the Born approximation is significantly reduced compared to the $\sigma =55$pm case in Fig. 4(b): the local RMS error averaged over 0.5m remains below 2.5 pm beyond 25m of AGF, and the large error spikes in Fig. 4(b) are not present.

One practical problem with this approach is that such AGFs are not currently available off-the-shelf. An alternative solution is to splice together AGF with different center wavelengths. For example, 10m of fiber with center wavelength 1550nm can be spliced to 10m of fiber with center wavelength of 1549nm to obtain an AGF with 20m sensing range. The backreflected light from the 1549nm fiber will not be affected by 1550nm gratings, unless those gratings are sufficiently strained to shift their center wavelength by about 1nm. A shift of 1nm over a significant enough length of fiber to cause Born approximation errors seems unlikely. AGF fibers with different center wavelengths are commercially available, so this may be a useful way to extend the range of AGF distributed strain sensing.

We have performed a pair of strain measurements to test this idea. First, we applied approximately constant strain to a 5m section of a 20m long AGF with a center wavelength of 1550nm. The strain measurement is shown in Fig. 6(a)). The same measurement is made with a fiber made of 10m of AGF with a 1550nm center wavelength spliced onto 10m of AGF with a 1549nm center wavelength. All AGF fibers used here have $\sigma =55$ pm. The result is shown in Fig. 6(b)). We note that the 1550nm AGF appears to have increasing noise at longer distances, including a large error spike of about 50pm occurring near $z=19$m. The hybrid 1550nm-1549nm AGF has a constant noise level down the fiber and has no large error spikes.

 

Fig. 6. Increasing the range of OFDR strain measurements by using AGF with different center wavelengths. The standard deviation of the center wavelength for all AGF was $\sigma =55pm$. a) Measurement of constant strain applied to 5m of a 20m AGF with 1550nm center wavelength. b) Measurement of constant strain applied to 5m of a 10m of AGF with 1550nm center wavelength spliced to 10m of AGF with 1549nm center wavelength. c) Zoom in of Fig. 6(a) (gray) and RMS error taken over a 0.5m window (black). he RMS error in the 20m of AGF centered at 1550nm increases from about 4pm to about 6pm, before suddenly jumping to 10pm due to the large (50pm) error spike. d) Zoom in of Fig. 6(b) (gray) and RMS error taken over a 0.5m window (black). The RMS error of the 1550nm-1549nm hybrid AGF is constant at about 4pm, with no large error spikes

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We quantify the difference by computing the RMS error averaged over 0.5m of fiber. These results are shown by the black line in Figs. 6(c)) (1550nm AGF only) and d) (1550nm-1549nm hybrid). The RMS error in the 20m of AGF centered at 1550nm increases from about 4pm to about 6pm, before suddenly jumping to 10pm due to the large 50pm error spike, while the RMS error of the 1550nm-1549nm hybrid AGF is constant at about 4pm, with no large error spikes.

4. Conclusions

We have investigated the limits of two different optical fiber types for distributed strain sensing using OFDR. We have developed a model for AGF and performed a detailed investigation of the measurement errors due to commonly used approximations in OFDR distributed optical fiber sensing with all-grating fiber. Our model predictions have been confirmed by measurements using a commercially available OFDR. A key result is that the oft-neglected autocorrelation term can limit the sensing range for $q(z)$ measurements when using AGF; this limitation seems to be less problematic when performing distributed strain measurements using the “window-FFT-xcorr” technique.

We also find that the sensing range of AGF, typically limited by multiple reflections in the grating, can be significantly improved by increasing the spread in center wavelengths of the weak gratings in the AGF. Further, we demonstrate that the range of AGF measurements can be extended by concatenating segments of commercially available AGF with different center wavelengths. Our results show how to extend the sensing range of OFDR using AGF, which may be especially useful for applications such as structural health monitoring in which high spatial resolution, high measurement speed, and low processing costs are desired.