Abstract

Monitoring strain is important in precision engineering applications that require maintaining the precise alignments of structures over time, such as those found in machine tools and metrology frames. We present a fiber-optic strain measurement technique based upon broadband interferometry that is variously configurable in terms of gauge length and sensitivity. This is achieved by the use of an unbalanced interrogation interferometer configuration that alleviates the cavity length limit imposed by the temporal coherence of the system. We also demonstrate that dispersion in fiber sensors based on intrinsic Fabry–Perot interferometers must be considered to optimize performance. The possibility of multisensor interrogation using the same optical system is also reported.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

1. INTRODUCTION

Strain monitoring is a necessary activity in many areas of precision engineering, where an understanding of structural deformation due to loading and thermal effects [1] is required to optimize performance, for instance, in the characterization and compensation of thermal errors in machine tools [2]. Strain sensors based on the propagation of light in optical fiber have a number of distinct advantages over electrical strain gauges, such as immunity to electromagnetic noise, greater longevity in corrosive environments, easier mounting/integration, and, in some embodiments, improved absolute strain sensitivity. Here we report on a strain sensing approach based on broadband interferometric interrogation of an optical fiber cavity that has the potential to yield high-strain resolution without the need for expensive swept laser sources.

A. Strain Measurement with Optical Fibers

Fiber Bragg gratings (FBGs) are currently the most common method of measuring strain using optical fiber. They are an example of an intrinsic all-fiber sensor in that the construction of the active element is monolithic and relatively easy to produce using a femtosecond laser inscription process [3]. The interrogation method requires the determination of the wavelength of a narrow peak in the spectrum of light reflected from the FBG, which is strain-dependent. However, strain sensitivity is typically limited to about a microstrain per picometer (in wavelength). For commercial systems, this imposes a strain resolution limitation in the region of 1 µɛ due to the challenges of providing high-resolution spectral detection economically. Swept-source laser systems are another approach for increasing interrogation resolution, but these are still not cost-effective enough for many potential applications. For applications involving longer time-scale measurement, maintaining wavelength stability of sources/detectors can pose an added challenge.

For higher resolution strain monitoring, a common approach is to harness the inherent sensitivity of interferometry, which can be embodied using fiber in combination with a Fabry–Perot cavity which acts as the sensing element; such approaches can potentially yield nanostrain sensitivity. Fabry–Perot cavity-based methods can be categorized by the nature of the active sensing element: extrinsic Fabry–Perot interferometers (EFPIs), where light is coupled out of the fiber into an external cavity typically use an air gap; and intrinsic Fabry–Perot interferometers (IFPIs), where the cavity is created by creating reflective locations in the fiber itself, for instance by inscribing a pair of Bragg gratings [4], or through the modulation of the core geometry [5]. Both IFPIs and EFPIs interrogate the extension of the cavity length under strain and a range of interferometric techniques can be applied to extract the resulting optical path length (OPL) change.

 

Fig. 1. Schematic of the optical apparatus. Light from an SLED is partially reflected from the fiber strain sensor formed from surfaces at R1 and R2. Returning light is split by an unbalanced Michelson interferometer, enabling the reflected light to interfere coherently before entering into a spectrometer.

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EFPIs have some benefits in terms of a low inherent temperature sensitivity due to use of an air gap. However, strain sensitivity is limited by the finite length of the air gap, which is typically limited to a few tens of micrometers before fringe visibility losses become a limitation [6]. More complex and costly constructions can alleviate this limit, using, for instance, the fusion splicing of a graded-index (GRIN) collimator within the cavity [7] EFPIs. These more complex multielement constructions are not suited to many applications due to concerns associated with alignment robustness, tensile strength, cost, and overall sensor footprint. For some specialist applications, e.g., in vacuum, other factors such as possible outgassing from adhesives required for construction may also be a consideration. IFPI construction requires more specialist equipment, e.g., for femtosecond laser inscription or micromachining, but these methods can be low cost when scaled up in a similar manner to commercial FBG production. IFPIs do not possess such cavity length limitations as EFPIs because the light remains confined in the fiber throughout, so cavity length can be extended in order to improve sensitivity. However, the coherence length set by the source/detection bandwidths still sets a limit on the overall cavity length in many systems. In EPFI sensors employing broadband detection, the adverse effects of dispersion are minimized because the cavity is generally an air gap. Conversely, for IFPIs, the impact of waveguide dispersion must be considered, and possibly corrected for, if optimal sensor performance is to be realized. This becomes more important as larger cavity length is utilized to realize high sensitivities. In this paper, we introduce an IFPI sensor employing a broadband interrogation system based around an unbalanced Michelson interferometer configuration. This configuration alleviates the cavity length limit imposed by the temporal coherence of the system, allowing for an arbitrary sensor length to be employed in order to tailor sensitivity for a specific application. We also consider the impact of waveguide dispersion on the performance of the sensor and give recommendations for optimal design parameters to minimize these systematic effects. The broadband interrogation method provides absolute strain measurement, which is important for applications involving relatively long time scales or where measurement must be maintained after power loss, optical disconnection of interrogation apparatus, or technical problems. The construction of our sensor is simple, and the overall system is relatively simple, requiring a low-cost superluminescent diode (SLED) source and basic spectrometer.

The paper is structured in the following manner: Section 2 describes the construction of the specific broadband IFPI apparatus investigated, consideration of the impact of systematic dispersion, and an overview of the optical signal generation and analysis; Section 3 covers the calibration process applied through a specifically design sensor evaluation rig; Section 4 discusses briefly the potential of multisensor interrogation using this sensor; and Section 5 summarizes the findings and gives consideration to further investigation.

2. EXPERIMENTAL DETAILS

A. Optical Apparatus

Figure 1 shows a schematic representation of the optical apparatus investigated, which comprises a remote sensor linked by optical fiber to an interrogation interferometer. A broadband fiber-coupled SLED source illuminates the fiber strain sensor, with returning light traversing a Michelson interferometer before being detected by a spectrometer. The following subsections break down the operational details of specific elements.

1. Light Source and Fiber Strain Sensor

The experimental setup for the strain sensor is shown in Fig. 1. Light from a fiber-coupled SLED (Exalos EXS0850-055, FWHM linewidth $\sim{55}\;{\rm nm}$ centered at 855 nm) exits the interrogation interferometer and traverses a single-mode fiber patch cable of configurable length. It then enters into the remotely located strain sensor created from a section of single-mode optical fiber (Nufern 780HP) of length $ d $ (where $d = 150\; {\rm mm}$ in this instance) that has reflective surfaces at either end created by vacuum-sputtered titanium oxide and silicone dioxide layers (${\rm Ti}_3{\rm O}_5/{\rm Sio}_2$). The first reflector, R1, is deposited onto a polished LC/PC ferrule (F2) and reflects approximately 30% of the light. F2 is mechanically coupled to another ferrule, F1 using a split ceramic sleeve (Thorlabs ADAL1). The other end of the strain-sensitive fiber sensor is terminated by a cleaved fiber face coated with a 100% reflective coating to form the second reflector (R2). The ratio of R1/R2 reflectance is chosen so that upon return to the interrogation interferometer, the intensity of each arm of the interferometer is approximately equal, which yields optimal fringe visibility in the retrieved spectral interferogram. This construction process, while not at this stage strictly monolithic, allows for the rapid development of the method and achieves sufficient miniaturization for the proof-of-concept tests described in this paper. Future work will take advantage of the increased miniaturization and robustness afforded by previously reported IFPI construction techniques to yield a truly monolithic all-fiber sensor construction [8].

2. Method for Alleviating Temporal Coherence Limits on Cavity Length

Cavity length is an important parameter for interferometric strain sensors because it determines the sensitivity; however, as pointed out in Section 1.A, there is often a limit to the maximum cavity limit that may be achieved in IFPIs due to the temporal coherence of the light source when broadband interrogation is used. The geometric length separation between the reflectors in the fiber sensor is approximately 150 mm in our setup. Such a large separation between the reflective surfaces (R1 and R2) would normally require substantial temporal coherence to generate an interference signal if the beams were to be combined directly. To mitigate this issue, we use an additional pair of mirrors (M1 and M2; see Fig. 1) to form an interrogation interferometer that reduces the optical path difference between a portion of the light from each reflector before it enters the spectrometer. This allow coherent interference to take place but at the expense of an increase in incoherent background signal. Furthermore, it enables the path imbalance to be tuned so that our signal-processing method can operate optimally for any given sensor cavity length.

The light returning from the strain sensor is a combination of light that has reflected from R1 and R2, ignoring any multiple reflections for the sake of simplicity. Within the interrogation interferometer, this light is first collimated into free space prior to being split into two orthogonal beams by a nonpolarizing 50/50 beam splitter. One arm reflects from M1 at a distance ${a_1}$ away, the other arm from M2, at a distance ${a_2}$ away. At the point where the light initially meets the beam splitter (i.e., after collimation from the fiber), the light reflected by R1 has propagated through a total OPL of ${O_1}\{k \}$, while the light reflecting from R2 has propagated through an OPL of ${O_2}\{k \}$, with $k = 2\pi /\lambda$ being the wavenumber. When returning to the beam splitter, the light from M1 and M2 has propagated through an additional OPL of $2{n_1}{a_1}$ and $2{n_2}{a_2}$, respectively. As such, the light entering the spectrometer is a combination of light that has propagated through four distinct OPLs, made up from reflected light from R1 or R2 prior to entering free space, and reflecting light from either ${M_1}$ or ${M_2}$ after the beam splitter. We will denote the OPL of each of these routes as ${\rm OPL}_{p,q}$, with $p = 1,2$ corresponding to light reflecting from ${R_1}$ or ${R_2}$, respectively, and $q = 1,2$ corresponding to reflections from ${M_1}$ or ${M_2}$. By carefully selecting the values of ${a_1}$ and ${a_2}$, the OPLs of ${{\rm OPL}_{1,1}} = {O_1}\{k \} + 2{n_1}{a_1}$ and ${{\rm OPL}_{2,2}} = {O_2}\{k \} + 2{n_1}{a_2}$, can be made approximately equal. We state approximately here as the OPL is wavenumber-dependent in the fiber, but approximately wavelength-independent in air; thus, it is strictly only possible to equalize them at a single wavelength. Nonetheless, the OPLs can be matched sufficiently across all wavenumbers so that temporal coherence can be maintained sufficiently to generate interference.

There are also the two other OPLs that the light may propagate through (${{\rm OPL}_{2,1}}$ and ${{\rm OPL}_{1,2}}$). These are incoherent contributors, which add only to the background signal, because the OPL differences are beyond the coherence length of the detection system. For the same reason, we ignore any light that has undergone multiple reflections in the probe, considering this as a contributor to the background signal only.

3. Spectrometer

After being recombined at the beam splitter, the light passes into a spectrometer created from a diffraction grating, spherical mirror, and CMOS line array camera (Basler Racer raL8192-gm). Analyzing the light in this manner, we can consider the result from the line array as being equivalent to a set of (nearly) monochromatic systems, all taking their results in parallel, but at different wavelengths. With this apparatus, we have achieved a sampling rate of 6 kHz, which is currently limited by the speed of the camera.

B. Signal Generation

1. Phase Variation in Air versus in Fiber

In some regions of the system, the light is propagating within an optical fiber, while in other regions, the medium is air; these differences must be carefully considered when analyzing the signal. In the regions where the light is propagating in air, we can consider the light to be well approximated by a plane-type wave, $E = {\rm Re}\{{E_n}\exp ({i({kx - \omega t})})\}$, where ${E_n}$ is the magnitude of the plane wave. In the following analysis, we will not be overly concerned with the theoretical values of the magnitudes of each of the waves at any particular point in the system; the primary concern is the phase of the light along two distinct paths from the light source to the detector: ${P_{1,1}}$ from the light source to ${R_1}$, and then along the longer arm after the beam splitter, ${a_1}$; ${P_{2,2}}$ from the light source to ${R_2}$, and then along the shorter arm after the beam splitter, ${a_2}$. Here, ${P_{1,1}}$ and ${P_{2,2}}$ are related to ${{\rm OPL}_{1,1}}$ and ${{\rm OPL}_{2,2}}$, respectively, but also include any additional common path elements from the light source through to the spectrometer not covered by the definition in Section 2.A.2.

 

Fig. 2. Plots of the normalized propagation constant, $b$, against the normalized frequency, $v$, in a typical optical fiber. The space between the vertical (red) lines in (a) indicates the normalized frequency range covered by the SLD light source used. A plot of the normalized propagation constant within this range (b) shows a close to linear relationship with the normalized frequency within this range.

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We are most interested in the phase information because this is primarily determined by the length of the fiber sensor cavity, which is dependent on any imparted strain. The phase will describe the form of the interference signal that is generated. The relative magnitudes of the interfering light will determine fringe visibility, which we are less concerned with, as a process of signal normalization is undertaken in our analysis method.

Within the optical-fiber sensor, consideration of light propagation is more complex than in free space, because it is not solely the wavenumber that will give us the phase advancement, but rather the propagation constant, $\beta$ [9]. While we do not know the exact details for the optical fiber we are using, as precise designs details are carefully guarded commercial intellectual property (IP), we can look at the general behavior of optical fibers to inform us about what to expect. Furthermore, we will show that this is borne out by the experimental results discussed in Section 3.

Figure 2 shows a plot of the normalized propagation constant, $b$, against the normalized frequency, $v$. Here $v = rk({n_c^2 - {n^2}})$, where $r$ is the fiber-core radius, and ${n_c}$ and $n$ are the refractive index of the core and cladding, respectively. While the assumption that the refractive index is constant in the core may not be strictly correct for all fiber designs, this will still indicate the general behavior. The normalized propagation constant is $b = 1 - {u^2}/{v^2}$, where $u = ({1 + \sqrt 2})v/({1 + {{({4 + {v^4}})}^{1/4}}})$ [9]. The normalized propagation constant does not vary linearly with normalized frequency in general; thus, if we double $k$ (and hence our value of $v$), we do not simply halve the wavelength in the direction of propagation, as would be the case of a plane wave in air. Figure 2(a) illustrates this by plotting $b$ against $v$, calculated by using the known core radius of the fiber in our strain sensor ($r = 2.2 \times {10^{- 6}}$) and assuming typical index values for a silica fiber, ${n_c} = 1.4475$ and $n = 1.4417$.

The range of normalized frequencies that correspond to the wavelength range of the superluminescent diode (SLD) illuminating our strain sensor (827.5–882.5 nm) is indicated by the two red vertical lines in Fig. 2(a). Figure 2(b) shows this portion and indicates that over this more limited range the relationship between $v$ and $b$ is well approximated by a linear relationship. However, it is clear that should a broader linewidth source or different illumination wavelength range be used, the signal-analysis methods we illustrate in the next section may need to be adjusted to account for nonlinearity. In this analysis, we have not considered the impact of fiber-core deformation because the fiber stretches with applied strain (i.e., the radial profile remains constant). This is a valid approximation if the unstretched core of the fiber used is considered to be a cylinder of radius of radius 2.2 µm and length 150 mm. A strain-induced length change of 1 mm would yield a core radius reduction of 7.3 nm if the overall volume is maintained. We have not at this time considered elasto-optic effects in this work, as the linearity of the unwrapped phase generated (see Fig. 4) suggests that these do not make a significant contribution in the system described.

2. Signal on the Detector

As mentioned in the previous section, in order to calculate the signal that is seen on the detector, we need to consider the phase difference between light propagating along two distinct paths, ${P_{1,1}}$ and ${P_{2,2}}$. Light following any other route is assumed to be combine incoherently and to add to the background signal. When exiting the fiber, the phase difference between light reflecting from R1 and R2 is $2\beta d$, and upon returning to the beam splitter after reflecting from M1 and M2, it is $2\beta d + 2k{a_2} - 2k{a_1}$. The phase difference is thus $\phi = 2\beta d + 2k({{a_2} - {a_1}})$.

The spectrometer pixels record the light at an equally spaced set of wavenumbers from ${k_{{\min }}}$ to ${k_{{\max }}}$, with the signal on each pixel being of the form of a superposition of two waves with a phase difference $\phi$ between them [10]. Thus, on each pixel the detected intensity is

$$\begin{split}I\{k\} &= \left[{{E_0}\{{k} \} + {E}_1\{k\}{\exp} ({i\phi \{k\}} )} \right]\\&\quad\times\left[{{E_0}\{k\} + {E_1}\{k\}{\exp} ({- i\phi \{k\}} )} \right] + {\rm D.C}. \\ &= E_0^2\{k\} + E_1^2\{k\} + 2{E_0}\{k\}{E_1}\{k\}\cos (\phi \{k\}) + {\rm D.C.}\end{split}$$
Here, ${E_0}$ and ${E_1}$ are the magnitude of the coherent light that interferes after following paths ${P_{1,1}}$ and ${P_{2,2}}$, respectively. All of the incoherently combining light in the system is represented by the D.C. term. If there is a linear relation between the propagation constant and the frequency of the light such that $\beta = \gamma k$, then $\phi = 2\beta d + 2k({{a_2} - {a_1}}) = 2k({\gamma d + ({{a_2} - {a_1}})})$. This approximation is justified over the limited range of wavenumbers that illuminate our system [see Fig. 2(b)]. A plot of $I\{k \}$ against $k$, as is provided by the spectrometer, will yield a sinusoidal signal over a fixed background [11] as $\phi$ changes linearly with $k$. The frequency of the sinusoid will be dependent on $\gamma d + {a_2} - {a_1}$. If ${a_1}$, ${a_2}$, and $\gamma$ are constant, then any change in the signal will be related solely to changes in $d$. Thus, a change in frequency of the sinusoid on the detector can be related directly to a change in the length of the fiber as the result of an applied strain.
 

Fig. 3. Analysis of a typical signal generated by the optical apparatus. The line between the upper and lower envelopes (red) in (a) shows a typical signal recorded by the spectrometer, with the lines above and below (blue) indicating the envelope on the interference fringes. The signal after the background signal and the envelope have been removed is shown in (b), where the underlying sinusoidal signal of interest is clearly present.

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Fig. 4. Part (a) shows the unwrapped phase (blue line) and a best-fit straight line (dashed line, red). Part (b) shows the difference (residual) between these lines.

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Figure 3(a) shows the typical form of a signal recorded by the spectrometer in our apparatus (red line), and it is apparent that a significant background signal is present. Here, the varying signal magnitude is primarily influenced by the spectral profile of the source. A process of normalization is undertaken to help extract the underlying sinusoidal signal. This is achieved by recording the maximum and minimum intensity values on each pixel as the fiber is stretched because the maximum value is limited by the upper envelope and the minimum by the lower envelope. The envelopes enclosing the signal (blue lines) in Fig. 3(a). The signal is normalized by subtracting the lower envelope, dividing by the difference between the upper and lower envelopes, and finally subtracting the mean.

 

Fig. 5. (a) shows a schematic of the apparatus used to apply strain to the fiber sensor while simultaneously monitoring the change extension using an external interferometer. (b) shows a computer-aided design (CAD) model of the 3D printed fixture utilized for this purpose.

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The first step is to isolate the region where the signal is strongest (between pixels 3000 and 6500) in this example and removing the effect of the envelope. This is done by subtracting the lower envelope, dividing by the difference between the lower and upper envelopes, and then subtracting the mean value). A low-pass filter is then applied to limit noise, which leads to a signal of the form shown in Fig. 3(b). Now most of the signal due to the fixed terms in Eq. (1) has been removed, and what is left is a scaled form of the $\cos (\phi)$ term. To recover $\phi$ from this signal, we apply a well-established approach related to that described by Takeda et al. [12]. This involves Fourier-transforming the signal; setting the negative frequency components to zero; inverse Fourier transforming; and finally, calculating the phase angle of the polar form of the complex-valued results. Figure 4(a) shows the unwrapped phase (blue) along with a best fit straight-line dashed line (red), while Fig. 4(b) plots the residual error between the two. For exactness, this phase is labeled $\theta$ instead of $\phi$, where $\theta = \phi + 2n\pi$. Strictly speaking, we obtain $\phi ({\rm mod}2{\pi})$ but nonetheless, unwrapping the phase and determining the gradient provides all the information required without having to determine $n$. The fact that the instantaneous phase in the signal increases linearly, corresponding to a sinusoid of fixed frequency, shows the previously stated approximation that $\beta$ varies linearly with $k$ is justified. If this were not the case, the period of the detected sinusoid would appear to vary.

In order to relate a change in frequency of the detected sinusoid to a strain-induced change in sensor length, $d$, it is necessary to determine the scaling factor, $\gamma$. To calculate this directly, it is necessary to know specific parameters for the optical fiber, e.g., precise refractive index profile of the core/cladding. Since such proprietary information is not readily available to end users, in the following section we present an apparatus to apply a well-controlled strain to our sensor in order to determine $\gamma$ empirically.

3. CALIBRATION

A. Sensor Extension Apparatus

A 3D printed fixture (Fig. 5) was created to allow the fiber sensor to be strained repeatedly and in a well-controlled manner. The fiber sensor was glued into a channel that runs along the length of the fixture, and two piezoelectric translator (PZT) actuators are used apply strain to the structure. The resulting extension is measured simultaneously by both the fiber sensor and a commercial displacement measuring interferometer (Renishaw ML10) featuring a measurement resolution of 1 nm, and quoted accuracy of $\pm 0.7\; \unicode{x00B5}{\rm m}/{\rm m}$ is used as a trusted reference. To maximize stability, the fixture was mounted upon the granite bed of a coordinate measurement machine (CMM).

The beamline of the ML10 interferometer is collinear with the fiber sensor and as close to coaxial as physically possible to minimize Abbe error. Light from the ML10 is incident upon a beam splitter, C1 is placed above the location of the first reflecting surface in the fiber sensor (R1) and divides the beam into a fixed reference beam, and a measurement beam that is returned by corner cube, C2, is mounted above the reflector at the end of the fiber, R2. The corner cube moves collinearly with the fiber end, so any measured change in the distance between C1 and C2 is ideally equivalent to the change in the length of the fiber sensor. The ML10 reference interferometer has a resolution of 1 nm and is capable of relative measurement at a 40 kHz sampled rate.

B. Calibration Results

Section 2.B.2 shows that as the fiber sensor cavity length, $d$, changes, the period of the sinusoid should also change (which corresponds to a change in the gradient of the instantaneous phase with respect to spectrometer pixel number). We noted above that $\phi = 2k({\gamma d + ({{a_2} - {a_1}})})$, and thus to determine $d$ from $\phi$, we would also need to know ${a_1}$ and ${a_2}$. However, to monitor strain, it is not the absolute length of $d$ that needs to be determined, but instead the change in $d$.

Making the substitution $d = {d_0} + {\Delta}d$ yields $\phi = 2k({\gamma {d_0} + ({{a_2} - {a_1}})}) + 2k\gamma {\Delta}d$. By setting the value of ${d_0} = - \frac{{({{a_2} - {a_1}})}}{\gamma}$, it can be seen that $\phi = 2k\gamma {\Delta}d$, and

$$\frac{{d\phi}}{dk} = 2\gamma {\Delta d}.$$
Thus, the gradient is related to the difference between $d$ and some value ${d_0}$. Determining a value for $\gamma$ would need knowledge of the actual wavenumbers recorded on each spectrometer pixel, but the same result can be obtained by calculating the gradient of $\theta$ with respect to pixel number and combining all the constants together to yield $d\phi /d({{\rm pixel}\,{\rm number}}) = S{\Delta}d$, assuming the wavenumber of the light detected on each spectrometer pixel varies linearly. The value for $S$ can be determined by a comparison of the change in gradient recorded by the strain sensor and the change recorded by the ML10 reference interferometer.

Figure 6(a) shows the change in the distance between C1 and C2, recorded by the ML10 as the fiber sensor is extended through 10 increments of approximately $1.6\; {\unicode{x00B5}{\rm m}}$. Figure 6(b) shows the calculated gradient of $\theta$ with respect to pixel number at each measurement position as calculated by the method introduced in Section 2.B.2. The ML10 is recording data at a higher rate than the fiber strain sensor, and so better captures details about the transient behavior after a step is taken; for instance, mechanical oscillation is apparent as the system settles after each increment in Fig. 6(a). Slightly more noise is also apparent from the fiber sensor, the cause of which will be addressed in later work as the system is better optimized. Nonetheless, the results obtained from the two measurement systems are highly comparable. In order to calculate $S$, the 10-increment stepping motion was repeated nine times. For each motion increment, recorded data (after mechanical settling) was averaged to reduce noise and yield a single value for that increment. The values of the distance (as measured by the ML10 interferometer) and the gradient (derived from the fiber sensor) at the first and last step were used to calculate a value of $S$ for each run. Across all nine runs, a mean value of $S = 9.7089 \times {10^{- 3}}$ with a standard deviation of $0.0411 \times {10^{- 3}}$ was obtained.

 

Fig. 6. (a) shows the change in length of the fiber as recorded by the ML10 interferometer as the fiber sensor is extended in 10 steps each of ${\sim{1.6}\;{\unicode{x00B5}{\rm m}}}$. (b) shows the corresponding changes in the gradient of the unwrapped phase, $ \theta $.

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Fig. 7. Change in the length of the fiber as recorded by the strain sensor plotted against the change recorded by the ML10 interferometer. Each point is the average of nine runs, with horizontal and vertical error bars indicating the minimum and maximum value that was recorded for any of the runs at that point. The green line is a best-fit straight line to the data points.

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Figure 7 shows the results of the nine repeated stepwise extensions of the fiber, plotting sensor extension against the displacement as recorded by the ML10 interferometer. Each of the plotted points on the $x$ axis is the mean displacement recorded by the ML10 over all nine runs. On the $y$ axis, each point is the mean of the nine calculated sensor extensions. The error bars correspond to the minimum/maximum range across all nine runs at each increment of applied strain; the greatest difference between the maximum and minimum position recorded by the strain sensor across all of the points is 0.2165 µm. This will comprise contributions from the repeatability of the strain sensor and the repeatability of the rig. We anticipate that with a rig optimized for mechanical performance, environmental control, and stable strain-sensor mounting, that the achievable repeatability will far exceed this.

The solid line (green) is a best-fit straight line to all of the measured data points. In a perfectly calibrated system that was free of motion errors, all of the points would lie on a line having a gradient of 1. In fact, the best-fit line has a gradient of 1.005 and a nonzero intercept (${-}4.03 \times {10^{- 5}}$) and the points oscillate slightly above and below. The most likely explanation for this is that the reflector on the end of the strain sensor and the corner cube reflector for the ML10 are not moving perfectly collinearly, due some combination of misalignment and distortion of the 3D printed fixture. Nonetheless, the correspondence is quite reasonable and implies that the strain sensor can detect changes in length quite reliably. Further improvements to this apparatus could be made to enable a better calibration, and hence more accurate determination of $S$ to be achieved. A high priority in this regard would be to improve the structural rigidity of the 3D printed fixture.

4. MULTISENSOR INTERROGATION

With the technique described in this paper, it is possible to multiplex several IFPI fiber sensors and interrogate them simultaneously using spatial frequency division multiplexing. Because each fiber sensor generates a sinusoidal signal of the form shown in Fig. 3, it is possible to separate several signals in the Fourier domain during the initial filtering process. To achieve this, the unstrained cavity length of each sensor must be sufficiently different that the spatial frequencies generated in the Fourier domain can be separated, after which the analysis for each sensor can be carried out independently. The upper limit to the number of sensors that can be multiplexed is a topic of further work and will depend on several factors, including filtering parameters, effect of cross talk, and the operating range of the individual sensors. Furthermore, the range of lengths over which the sensor cavities can be constructed is limited by the coherence length enforced by the detection, as each sensor must operate within a subset of the overall system bandwidth.

5. SUMMARY AND CONCLUSIONS

This paper reports the demonstration of a broadband interferometric interrogation method for absolute strain measurement implemented using an IFPI fiber sensor. The sensor length is not limited by the coherence of the broadband light, because the system features an interrogation interferometer that balances path lengths of light returning from the sensor. This means that the technique is applicable to both short sensors, where strain measurement is highly localized, and longer ones, where higher sensitivity can be achieved. In addition, we have considered the effect of waveguide dispersion on the signal analysis and determined this empirically. Critical to the effective operation of the sensor is that the illumination wavenumber range lies within a region of operation where the fiber propagation constant varies linearly with wavenumber. However, correction for systematic dispersion may still be necessary in very high-precision strain measurement applications. The severity of any nonlinearity in the strain response and its subsequent correction will be a topic for further work. Also highlighted is the potential for interrogating several sensors using a spatial frequency division multiplexing technique implemented by varying the cavity length of each individual fiber sensor to enable signal separation in the Fourier domain, and thus their independent interrogation. The relatively low cost of the apparatus, which does not require expensive swept-laser sources or high-resolution wavelength interrogation, is a distinct advantage. Further work is planned to more comprehensively determine the key metrological characteristics of the sensor and demonstrate its advantages for on-machine measurement of strain in high temperature and radiation environments.

Funding

Science and Technology Facilities Council (ST/N001699/1); Engineering and Physical Sciences Research Council (EP/P006930/1); High Luminosity Large Hadron Collider (HL-LHC) project.

Acknowledgment

The authors are thankful for the contributions made to experimental procedures by Mr. Ali Rahmatinia while undertaking the MSc by research at the University of Huddersfield.

Disclosures

The authors declare no conflicts of interest.

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2. J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012). [CrossRef]  

3. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997). [CrossRef]  

4. F. Shen, “UV-induced intrinsic Fabry-Perot interferometric sensors and their multiplexing for temperature and strain sensing,” Proc. SPIE 6174, 61740D (2006). [CrossRef]  

5. Z. Huang, “Intrinsic Fabry–Pérot fiber sensor for temperature and strain measurements,” IEEE Photon. Technol. Lett. 17, 2403–2405 (2005). [CrossRef]  

6. S. Pevec and D. Donlagic, “All-fiber, long-active-length Fabry-Perot strain sensor,” Opt. Express 19, 15641–15651 (2011). [CrossRef]  

7. Y. Zhang, “Fringe visibility enhanced extrinsic Fabry–Perot interferometer using a graded index fiber collimator,” IEEE Photon. J. 2, 469–481 (2010). [CrossRef]  

8. B. H. Lee, “Interferometric fiber optic sensors,” Sensors 12, 2467–2486 (2012). [CrossRef]  

9. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971). [CrossRef]  

10. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier,2013).

11. J. Calatroni, A. L. Guerrero, C. Sáinz, and R. Escalona, “Spectrally-resolved white-light interferometry as a profilometry tool,” Opt. Laser Technol. 28, 485–489 (1996). [CrossRef]  

12. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982). [CrossRef]  

References

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  1. J. Bryan, “International status of thermal error research (1990),” CIRP Ann. 39, 645–656 (1990).
    [Crossref]
  2. J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
    [Crossref]
  3. A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
    [Crossref]
  4. F. Shen, “UV-induced intrinsic Fabry-Perot interferometric sensors and their multiplexing for temperature and strain sensing,” Proc. SPIE 6174, 61740D (2006).
    [Crossref]
  5. Z. Huang, “Intrinsic Fabry–Pérot fiber sensor for temperature and strain measurements,” IEEE Photon. Technol. Lett. 17, 2403–2405 (2005).
    [Crossref]
  6. S. Pevec and D. Donlagic, “All-fiber, long-active-length Fabry-Perot strain sensor,” Opt. Express 19, 15641–15651 (2011).
    [Crossref]
  7. Y. Zhang, “Fringe visibility enhanced extrinsic Fabry–Perot interferometer using a graded index fiber collimator,” IEEE Photon. J. 2, 469–481 (2010).
    [Crossref]
  8. B. H. Lee, “Interferometric fiber optic sensors,” Sensors 12, 2467–2486 (2012).
    [Crossref]
  9. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref]
  10. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier,2013).
  11. J. Calatroni, A. L. Guerrero, C. Sáinz, and R. Escalona, “Spectrally-resolved white-light interferometry as a profilometry tool,” Opt. Laser Technol. 28, 485–489 (1996).
    [Crossref]
  12. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [Crossref]

2012 (2)

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

B. H. Lee, “Interferometric fiber optic sensors,” Sensors 12, 2467–2486 (2012).
[Crossref]

2011 (1)

2010 (1)

Y. Zhang, “Fringe visibility enhanced extrinsic Fabry–Perot interferometer using a graded index fiber collimator,” IEEE Photon. J. 2, 469–481 (2010).
[Crossref]

2006 (1)

F. Shen, “UV-induced intrinsic Fabry-Perot interferometric sensors and their multiplexing for temperature and strain sensing,” Proc. SPIE 6174, 61740D (2006).
[Crossref]

2005 (1)

Z. Huang, “Intrinsic Fabry–Pérot fiber sensor for temperature and strain measurements,” IEEE Photon. Technol. Lett. 17, 2403–2405 (2005).
[Crossref]

1997 (1)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

1996 (1)

J. Calatroni, A. L. Guerrero, C. Sáinz, and R. Escalona, “Spectrally-resolved white-light interferometry as a profilometry tool,” Opt. Laser Technol. 28, 485–489 (1996).
[Crossref]

1990 (1)

J. Bryan, “International status of thermal error research (1990),” CIRP Ann. 39, 645–656 (1990).
[Crossref]

1982 (1)

1971 (1)

Askins, C. G.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier,2013).

Brecher, C.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Bryan, J.

J. Bryan, “International status of thermal error research (1990),” CIRP Ann. 39, 645–656 (1990).
[Crossref]

Calatroni, J.

J. Calatroni, A. L. Guerrero, C. Sáinz, and R. Escalona, “Spectrally-resolved white-light interferometry as a profilometry tool,” Opt. Laser Technol. 28, 485–489 (1996).
[Crossref]

Davis, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

Donlagic, D.

Donmez, M. A.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Escalona, R.

J. Calatroni, A. L. Guerrero, C. Sáinz, and R. Escalona, “Spectrally-resolved white-light interferometry as a profilometry tool,” Opt. Laser Technol. 28, 485–489 (1996).
[Crossref]

Friebele, E. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

Gloge, D.

Guerrero, A. L.

J. Calatroni, A. L. Guerrero, C. Sáinz, and R. Escalona, “Spectrally-resolved white-light interferometry as a profilometry tool,” Opt. Laser Technol. 28, 485–489 (1996).
[Crossref]

Härtig, F.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Huang, Z.

Z. Huang, “Intrinsic Fabry–Pérot fiber sensor for temperature and strain measurements,” IEEE Photon. Technol. Lett. 17, 2403–2405 (2005).
[Crossref]

Ina, H.

Jedrzejewski, J.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Kersey, A. D.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

Knapp, W.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Kobayashi, S.

Koo, K. P.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

LeBlanc, M.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

Lee, B. H.

B. H. Lee, “Interferometric fiber optic sensors,” Sensors 12, 2467–2486 (2012).
[Crossref]

Mayr, J.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Moriwaki, T.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Patrick, H. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

Pevec, S.

Putnam, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

Sáinz, C.

J. Calatroni, A. L. Guerrero, C. Sáinz, and R. Escalona, “Spectrally-resolved white-light interferometry as a profilometry tool,” Opt. Laser Technol. 28, 485–489 (1996).
[Crossref]

Schmitt, R.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Shen, F.

F. Shen, “UV-induced intrinsic Fabry-Perot interferometric sensors and their multiplexing for temperature and strain sensing,” Proc. SPIE 6174, 61740D (2006).
[Crossref]

Shore, P.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Takeda, M.

Uhlmann, E.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Wegener, K.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Wendt, K.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier,2013).

Würz, T.

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

Zhang, Y.

Y. Zhang, “Fringe visibility enhanced extrinsic Fabry–Perot interferometer using a graded index fiber collimator,” IEEE Photon. J. 2, 469–481 (2010).
[Crossref]

Appl. Opt. (1)

CIRP Ann. (2)

J. Bryan, “International status of thermal error research (1990),” CIRP Ann. 39, 645–656 (1990).
[Crossref]

J. Mayr, J. Jedrzejewski, E. Uhlmann, M. A. Donmez, W. Knapp, F. Härtig, K. Wendt, T. Moriwaki, P. Shore, R. Schmitt, C. Brecher, T. Würz, and K. Wegener, “Thermal issues in machine tools,” CIRP Ann. 61, 771–791 (2012).
[Crossref]

IEEE Photon. J. (1)

Y. Zhang, “Fringe visibility enhanced extrinsic Fabry–Perot interferometer using a graded index fiber collimator,” IEEE Photon. J. 2, 469–481 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Z. Huang, “Intrinsic Fabry–Pérot fiber sensor for temperature and strain measurements,” IEEE Photon. Technol. Lett. 17, 2403–2405 (2005).
[Crossref]

J. Lightwave Technol. (1)

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, and E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Laser Technol. (1)

J. Calatroni, A. L. Guerrero, C. Sáinz, and R. Escalona, “Spectrally-resolved white-light interferometry as a profilometry tool,” Opt. Laser Technol. 28, 485–489 (1996).
[Crossref]

Proc. SPIE (1)

F. Shen, “UV-induced intrinsic Fabry-Perot interferometric sensors and their multiplexing for temperature and strain sensing,” Proc. SPIE 6174, 61740D (2006).
[Crossref]

Sensors (1)

B. H. Lee, “Interferometric fiber optic sensors,” Sensors 12, 2467–2486 (2012).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier,2013).

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Figures (7)

Fig. 1.
Fig. 1. Schematic of the optical apparatus. Light from an SLED is partially reflected from the fiber strain sensor formed from surfaces at R1 and R2. Returning light is split by an unbalanced Michelson interferometer, enabling the reflected light to interfere coherently before entering into a spectrometer.
Fig. 2.
Fig. 2. Plots of the normalized propagation constant, $b$, against the normalized frequency, $v$, in a typical optical fiber. The space between the vertical (red) lines in (a) indicates the normalized frequency range covered by the SLD light source used. A plot of the normalized propagation constant within this range (b) shows a close to linear relationship with the normalized frequency within this range.
Fig. 3.
Fig. 3. Analysis of a typical signal generated by the optical apparatus. The line between the upper and lower envelopes (red) in (a) shows a typical signal recorded by the spectrometer, with the lines above and below (blue) indicating the envelope on the interference fringes. The signal after the background signal and the envelope have been removed is shown in (b), where the underlying sinusoidal signal of interest is clearly present.
Fig. 4.
Fig. 4. Part (a) shows the unwrapped phase (blue line) and a best-fit straight line (dashed line, red). Part (b) shows the difference (residual) between these lines.
Fig. 5.
Fig. 5. (a) shows a schematic of the apparatus used to apply strain to the fiber sensor while simultaneously monitoring the change extension using an external interferometer. (b) shows a computer-aided design (CAD) model of the 3D printed fixture utilized for this purpose.
Fig. 6.
Fig. 6. (a) shows the change in length of the fiber as recorded by the ML10 interferometer as the fiber sensor is extended in 10 steps each of ${\sim{1.6}\;{\unicode{x00B5}{\rm m}}}$. (b) shows the corresponding changes in the gradient of the unwrapped phase, $ \theta $.
Fig. 7.
Fig. 7. Change in the length of the fiber as recorded by the strain sensor plotted against the change recorded by the ML10 interferometer. Each point is the average of nine runs, with horizontal and vertical error bars indicating the minimum and maximum value that was recorded for any of the runs at that point. The green line is a best-fit straight line to the data points.

Equations (2)

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I { k } = [ E 0 { k } + E 1 { k } exp ( i ϕ { k } ) ] × [ E 0 { k } + E 1 { k } exp ( i ϕ { k } ) ] + D . C . = E 0 2 { k } + E 1 2 { k } + 2 E 0 { k } E 1 { k } cos ( ϕ { k } ) + D . C .
d ϕ d k = 2 γ Δ d .