Optical low coherence reflectometry and fiber Bragg gratings written in small diameter (50 micrometer) optical fibers were used for measurements of non-homogenous internal strain fields inside an epoxy specimen with sub-grating length resolution. The results were compared with measurements using Fiber Bragg gratings in standard size (125 micrometer) single mode fibers and show that smaller fibers are less intrusive at stress heterogeneities.
©2010 Optical Society of America
From their discovery in 1978 by Hill et al. , fiber Bragg gratings (FBG) were successfully used as sensors for strain, temperature, pressure, humidity and various other quantities . Compared to traditional mechanical and electrical sensors, fiber gratings offer advantages such as being compact, lightweight, and small; they are immune to electromagnetic interference, have high temperature resistance and a long lifetime. For these reasons FBGs find various applications in structural health monitoring, non-destructive testing, and damage detection [3–5]. Usually, FBG sensors are used as point sensors to measure a certain parameter (strain, temperature etc) at a certain position along the fiber. In this case, the size of the grating (usually a few millimeters) determines the spatial resolution. Multiplexing several gratings, it is possible to realize a quasi-distributed sensor system. Depending on the required resolution, the size and the spacing between gratings are chosen.
Recent significant efforts were devoted to increasing the spatial resolution of the FBG sensors from millimeters to micrometers in scale. Among different techniques, the most successful were optical frequency domain reflectometry (OFDR) [6, 7], and optical low coherence reflectometry (OLCR) [8, 9]; both techniques are interferometric allowing for the measurement of phase and amplitude of the light reflected from the device under testing. The resolution of OLCR or OFDR is given either by the bandwidth of the light source, or the tuning range of the laser. The measurement range is limited by the travel range of the translation stage in the case of OLCR or by the line width of the tunable laser in the case of OFDR.
Over the last few years, OLCR proved to be a powerful tool for the determination of various properties of fiber Bragg gratings, such as position and length , refractive index amplitude and maximum reflectivity , impulse response , or dispersion . From the measured impulse response, it is possible to retrieve the grating’s complex coupling coefficient, q(z), through inverse scattering algorithms, such as layer-peeling proposed by Feced et al.  and Skaar et al . The grating reconstruction is limited in general to a grating transmission of about −20 dB, which corresponds to a grating strength-length-product of ~3 in the case of a homogeneous grating; this limitation is due to noise amplification in the inverse scattering process that depends exponentially on grating strength [14, 15]. The complex coupling coefficient contains information about the local grating period (local Bragg wavelength), and the refractive index modulation along the grating length. The observation of changes in local FBG parameters opened new possibilities for high spatial resolution grating characterization and distributed sensing [16, 17].
From the mechanical point of view, once embedded in the host material, the optical fiber represents an elastic inclusion. In the majority of the cases, FBG sensors use a standard single-mode fiber with ~8 μm core and 125 μm cladding size; thus, the fiber can induce defects inside the material and the fiber size could influence the measurement interpretation. For these reasons, efforts to produce smaller diameter fibers were undertaken by various manufacturers. In addition, smaller size fibers have smaller bending stress for the same bending radius, which could prolong the lifetime of the sensors. To date, there have not been many reported attempts to write FBG in small-diameter fibers, except from the group of Takeda [18–21] who used small-diameter (core/cladding/coating, 8/40/52 micrometers) fibers developed by Hitachi Cable Ltd. Using standard FBG interrogation techniques, they performed several experiments, including: detection of the edge delamination in carbon fiber reinforced plastic (CFPR) laminates  and detection of microscopic damages in composite laminates . Measuring the effect of thermal residual stress in the composite materials , it was found that FBGs in small-diameter fibers experienced slightly less induced birefringence than those in standard fibers for the same embedding conditions.
In this article we present high-spatial-resolution measurements of the axial strain distribution using FBGs written in small diameter optical fiber and embedded in an epoxy specimen. First, the OLCR experimental setup and specimen preparation are explained, the results of the measurements using small-diameter and standard fiber are then presented, and, finally, the effect of residual stress on fibers with different diameters is calculated.
2. Experimental setup
The scheme of the OLCR setup used to perform distributed measurements is shown in Fig. 1 . Light from the superluminescent diode (SLD) and the tunable laser (TL) are time multiplexed, and then split into the test and the reference arm using a 3 dB directional coupler. The superluminescent diode has a Gaussian spectral profile with 50 nm spectral full width at half maximum and a center wavelength of 1550 nm. The tunable laser provides a reference for the phase measurements and allows for the measurement of the grating’s reflection spectra. Light from the reference and test arm is reflected back and interferes inside the coupler (CPL) only if the length difference of the two arms is smaller than the coherence length Lc of the light source. By scanning the mirror in the reference arm, the FBG in the test arm is probed with a high spatial resolution given by half the coherence length (Lr = Lc /2 = 14.5 μm in the fiber) of the light source.
The actual spatial resolution is given by data processing and is typically 100 μm due to spline functions used for smoothing the coupling coefficient phase data before calculating the derivative, in Eq. (1). The light is detected using two photodiodes in a balanced detection scheme (BD) and a lock-in amplifier (LIA), which is set to the frequency of the piezoelectric phase modulator (PM). For every position of the translation stage, amplitude and phase difference between the Bragg reflected broadband light and the tunable laser light are measured. If the tunable laser is set to the Bragg wavelength peak, the phase difference is constant. The optical signal to noise ratio of the set-up is about 20 dB below the Rayleigh scattering limit in the optical fiber (−120 dB). More details on the general performance of our system can be found in reference .
The described phase-sensitive OLCR allows measuring the impulse response of the FBG [9, 16]. Time domain layer-peeling  is applied to obtain the complex coupling coefficient, q(z), where z is direction of light propagation and axial strain. The local Bragg wavelength, λB(z), is calculated using the following equation [16, 17]:22].
3. Results and discussion
3.1 Fiber Bragg gratings sensors
For the measurements of the distributed strain along the fiber inside the epoxy specimen, three fiber Bragg gratings were embedded. Each FBG was inserted in a different specimen. Firstly, an un-chirped commercial FBG without apodization written in standard SMF-28 single-mode fiber with core/cladding dimensions of 9/125 μm was used. The grating length was 20 mm with a peak reflectivity of 50%, which corresponds to a grating coupling-length product of 0.88 which is well below the upper limit for reconstruction. Secondly, FBGs written in the small-diameter fiber (Silitec SA) with core/cladding dimension of 11.6/50 μm were used. The fibers were hydrogen loaded before inscription using ArF excimer laser, and thermally annealed after fabrication. The gratings were 13 mm long with typical peak reflectivity of 50%. The Bragg wavelength of all gratings was around 1550 nm.
The OLCR amplitude signal SOLCR is related to the reflection amplitude, r, of a grating of length LR and strength, κ, as with . In fact if we assume a minimal OLCR grating signal of ≈-100 dB, that can be reconstructed, and an OLCR resolution of LR = 14.5 μm, we obtain a value for the grating coupling coefficient of κ = 0.6 m−1 (Δn ≈4 × 10−7). As the reconstruction limit is κ L = 3, a maximum grating length of up to 4.7 m would be possible. Such long gratings could be measured with tens of micrometer resolution in OLCR using long translation stages and multi pass configuration, or in OFDR due to the small laser linewidth and tens of nanometer tuning range. Soller et al. reported an OFDR with a resolution of 22 microns and a meter-long measurement range .
Before embedding the FBGs into the epoxy, the measurement of the stress sensitivity for standard and small-diameter fiber was performed. Calibrated (using precisely measured weights) axial loads were applied and the corresponding changes in the Bragg wavelength were obtained for both fibers as shown in Fig. 2 in terms of stress. The results for the standard fiber correspond well to the results reported in the literature . The small fiber showed lower sensitivity to the axial stress, which might be attributed to the fiber material composition (Young's modulus). Assuming similar pe = 0.22, and fiber diameters of 125 ± 0.3 and 49.6 ± 0.3 μm the corresponding Young's moduli are ~70 GPa and ~81 GPa, respectively.
3.2 Sample preparation
Both FBG types were embedded in rectangular epoxy samples (Fig. 3(a) ). Rectangular blocks of dimensions 10 × 10 × 40 mm have been fabricated using a mixture of two resins DER330© DER732© and the hardener DEH26© provided by the DOW chemical company. Firstly, the mixture, in weight proportion of 70:30:10 respectively, was stirred in a vacuum to eliminate the presence of bubbles, and secondly casted in a specially designed horizontal mould with the fiber in place (Fig. 3(b)). To prevent fiber fracture at the exit points of the mould and avoid resin leakage, thin rubber bands (RB) were placed horizontally on the right and left upper and lower parts of the mould. The fiber was manually positioned in the mould with the FBG near the edge of the mould to achieve non uniform strains with a small pretention held with tapes on planes (PL) extended from both sides of the mould.
Once the fiber was fixed in the correct position the mould was closed with a cap (CP) with two openings: one in which the resin is casted, and the other one guaranteed air evacuation. This mould configuration allowed for good fiber placement and excellent external surface finish. Since the resin was a low temperature curing type, the curing phase was performed at 30 °C for 24 h. After this step, the specimens were removed from the mould in order to reduce the formation of high residual strains due to the strong interaction with the mould’s walls during post-curing. The bare specimens were then subjected to a post curing phase at 70 °C for 9 h followed by natural cooling inside the oven.
3.3 Measurement of embedded gratings
After embedding the FBGs into the epoxy specimen, reflectivity and OLCR spectra (amplitude and phase) were measured and compared with the spectra before embedding. Figures 4 and 5 show the reflectivity with the maximum normalized to 0 dB (Fig. 4(a)) and the OLCR phase measurement (Fig. 4(b)) for the SMF-28 fiber and as an example for the small diameter fiber 2 (Fig. 5). All gratings indicated a noticeable shift to a lower wavelength due to a compressive strain induced by the epoxy shrinkage. In addition, there was spectral broadening and chirping as a typical signature of non-homogeneous strains. The FBGs written in the small-diameter fiber showed a smaller shift compared to FBG in SMF-28, which can be explained by the lower stress sensitivity (Fig. 2).
From the spectral measurements, it is possible to obtain information on the average strain (proportional to peak shift) along the grating, but it is not possible to assess the strain distribution itself. However, using the OLCR measurement, the strain distribution can be retrieved from the local Bragg wavelength distribution. The distributed strain can be calculated using Eqs. (1) and (2) and a wavelength-strain conversion factor of for all fibers.
Figure 6(a) shows strain distributions along the FBGs as a function of the position inside the sample; these residual strains are the result of the differences in the physical properties of the epoxy and the fibers [23, 24]. The strain distributions show a steep gradient near the sample edge and a plateau in the middle where the strains are uniform. The strain data from the small fiber 2 extend almost to the specimen entrance because the FBG was closer to the surface. The residual strain of ~5500 με is similar (~10% difference) for big and small fibers and approximately the same to the level reported elsewhere under similar conditions . A clear difference is observed between the SMF-28 and the small diameter fiber 2 at the specimen entrance where the stress changes are maximal; this difference is attributed to the fact that, from the stress analysis standpoint, the smaller diameter fiber is less intrusive and disturbs to a lesser extent the strain field near the edge of the specimen.
Residual strains in epoxies are commonly modeled as equivalent thermoelastic strains induced by an equivalent temperature change in the specimen. Thus, to better understand the experimental results and assess the influence of the fiber diameter, finite element simulations were performed. Two cylinders of 12 mm in diameter, each one with an axially located fiber of diameters 125 μm and 50 μm were considered with material properties equal to those of the material used in the experiments. Since the dimensions of the specimen’s cross section were large compared to the fiber diameter, the cylindrical geometry considered in the simulations of the specimen with the small diameter does not influence the strain evolution along the axis. For simplicity and to qualitatively determine the influence of the fiber diameter on the residual strains, a uniform equivalent temperature change of 52 °C was considered. This temperature was calculated on the basis of a thermal expansion coefficient for the matrix αm = 11 × 10−5 °C−1 , and that the strain matches the maximum experimentally observed value of 5750 με (Fig. 6a). Note that since the thermal expansion of the fiber is very small, as compared to that of the surrounding epoxy, it was considered negligible. Regarding the elastic properties the following values were considered: for the epoxy a Young modulus of Em = 2.35 GPa, measured experimentally, and a Poisson’s ratio, assumed equal to, νm = 0.38 were used. The corresponding values used for the fibers were: small fiber Ef = 81 GPa, SMF-28 Ef = 72 GPa, and a Poisson's ratio of νf = 0.16 for both fibers.
The results of the simulations shown in Fig. 6(b) indicate that the calculated distributions correspond qualitatively well to the experimental results. The strain in the middle of the specimen is independent of the fiber dimension; while at the entrance of the specimen, a steeper strain gradient along the small diameter fiber is calculated. The measured gradients are smaller than the simulated ones for both fibers; this is mainly attributed to the uniform equivalent temperature change applied along the specimen in the simulations.
We presented distributed measurements of internal strain fields using fiber Bragg gratings in fibers of different diameters and optical low-coherence reflectometry. A comparison of small-diameter and standard size fiber showed that the influence of the sample entrance is different for the two fiber types, but the level of the strain in the middle of the sample was almost identical. Small diameter fibers are promising candidates to replace standard size fibers for sensing application in the near future as they are less intrusive at stress heterogeneities.
The authors wish to acknowledge the financial support of the Swiss National Science foundation, grant number 124397/1, and seed-funds from the EPFL School of Engineering and the EPFL Space center. Aiping Luo acknowledges the financial support from the Sino-Swiss Exchange scholarship (2007-2008). Silitec SA Switzerland is acknowledged for providing the small-diameter fibers.
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