5.2        Characterization of a Fiber Bragg Grating under Diametric Loading

5.2.5       Study on the PM fiber strain anisotropy sensitivity

An important work has been performed to find the physical origin that could explained the experimental strain anisotropy sensitivity observed in the diametric loading of a PM fiber. A finite element model of the PM fiber has been realized in the LMAF laboratory by Federico Bosia. The simulation of diametric load has been conducted to retrieve the strain field at the fiber core where the FBG is located. Using the material parameters provided by the manufacturer, a very small strain anisotropy is observed at the core location (less than 5 % compared to the isotropic fiber case). Other material parameters have been found in the literature and applied to the simulation. The found anisotropy is more important and explains partially the transverse stress sensitivity anisotropy of the FBG in PM fibers. The results presented hereafter are partly based on the joined-papers written in collaboration with the LMAF group [5-7, 5-8]. Nevertheless, it remains open questions to completely describe the behavior of FBG written in PM fibers subjected to transverse stress and further studies are required.

a)   Finite element modelization

Due to the complex structure of the fiber in this case, finite-element-method (FEM) simulations need to be carried out to determine the strain distributions generated in the fiber core and derive numerical predictions to be compared with experimental measurements. In order to do this, the residual strains responsible for the initial birefringence of the fiber need to be estimated, and then the response to diametrical compression evaluated. The FEM simulations are performed using the I-DEAS code.

Fig. 5-27 Micrograph view of the PM fiber (a) and finite element mesh used for the simulations

Fig. 5-27a illustrates a micrograph of the PM-fiber section. The borosilicate bow ties are clearly visible. Based on this geometrical information and on manufacturer specifications, the 2-D FEM mesh is constructed (Fig. 5-27b). The diameter of the fiber is 125mm and that of the mode field is 9mm. The mesh correctly models the borosilicate bow ties (about 15x20mm) and the silica-glass core and cladding, and is refined in the central fiber-core region where strains are calculated [5-23]. A Young's modulus of E_B=67 GPa and a Poisson's ratio of n_B=0.17 are used for borosilicate (data provided by Fibercore). As mentioned previously and as indicated in Fig. 5-28a, the coordinate axes parallel and perpendicular to the bow ties are indicated as x' and y', whilst those parallel and perpendicular to the loading direction are indicated with x and y.

b)   Simulation of the natural birefringence and of the diametric load

The residual strains responsible for the birefringence are estimated assuming a linear elastic thermal loading problem. The approach is similar to that employed in [5-18]. Thermal expansion coefficients of a_B = 14x10^-7°C^-1 and a_G = 5.5x10^-7°C^-1 are used for borosilicate and silica glass, respectively (data provided by Fibercore). Simulations are performed using both plane-strain and plane-stress elements. The assumption that the resulting residual strains generated in this type of geometry are equal and opposite along the slow and fast axes, respectively, can thus be verified. The ratio between the two strains is found to be e_R,1/e_R,2 = -0.89, therefore the previous assumption can be modified and this correction, though small, accounted for in calculations.

a) b)

Fig. 5-28 Axes definition (a) and diametric load geometry (b)

Additionally, simulations are carried out to determine the strains generated in the fiber core by diametrical loads in the same loading range as that considered experimentally. The strains are determined as a function of loading angle g (Fig. 5-28b). Due to the structure of the PM fiber, some anisotropy is expected, i.e. loading in the direction of the x'-axis in Fig. 5-28b should give smaller e₁ strains than the e₂ strains obtained when loading in the y'-direction. This is indeed the case, however, due to the small mismatch between the elastic properties of silica and borosilicate, this effect is found to be negligible. A difference of 5% at most is obtained with the strains calculated in a homogeneous isotropic fiber with no bow ties, as is the case for the standard SM fiber.

Fig. 5-29 Bragg wavelength deviation for an applied diametric load at an angle g = 54° (a) and FBG diametric load sensitivity calculated by linear fit for different loading ranges (b)

Having calculated the strains in the fiber core as a function of loading angle g, it is possible to highlight the influence of the initial birefringence of the PM fiber on the sensor response to transversal loading. Using the approach illustrated earlier, the expected Bragg wavelength shifts are calculated as a function of applied diametrical load for various loading angles. Whilst the response is linear when loading is directed along one of the polarization axes (g =0° or g =90°), this is no longer true for all other loading angles. For example, results are plotted for g =54° in Fig. 5-29a : the nonlinearity in this case is evident. This is due to the load-dependent rotation, described by equation (5-24), of the principal axes with respect to the initial polarization axes. Thus, the response of a FBG sensor written in PM-fiber to transverse loads applied at an angle to the fast and slow axes is nonlinear, at least in the range where the strains due to loading are of the order of the residual strains generating the birefringence. This is also consistent with experimental results.

Due to this behavior, an error is introduced when an angular sensitivity per unit load is defined, as done in [5-13, 5-21], because the slope of the wavelength shift changes with the load. Figure Fig. 5-29b shows the numerically calculated sensitivities when the slopes are taken at P/l=1N/mm and P/l=6N/mm. In both cases, the sensitivities for the fast and slow axes are plotted. It is apparent that for increasing loads, a deviation from the expected sinusoidal behavior is obtained.

Furthermore, only a very small anisotropy is observed, i.e. the sensitivity is nearly identical when loading is directed along the slow axis at g =0° and fast axis at g =90°. This is not the behavior encountered experimentally. The experimental measurements indicate that in fact the fast axis is considerably less "sensitive" to diametrical compression, by a factor close to 2. These results are also obtained in similar experiments in the literature [5-21, 5-22]. The reasons for this mismatch between experimental and numerical results are thus far unclear. One possibility is a rather large uncertainty on material properties of borosilicate. For example, in references [5-18] and [5-13] a Young's modulus and Poisson's ratio of 50.8 GPa and 0.21 are used, respectively. These values differ considerably from those provided by the manufacturers of the fibers used in this study. Therefore, both sets of material parameters are used in FEM simulations, and results are compared.

Fig. 5-30 shows the experimentally measured and numerically calculated sensitivities as a function of the loading angle for P/l =1 N/mm. The numerical values are determined using both E_B=67 GPa and E_B=50 GPa. It is clear that a greater mismatch between the Young's moduli of silica and borosilicate improves the agreement between experimental and simulated results, however, a considerable discrepancy remains. Other possible explanations for this discrepancy are an oversimplified model for the loading configuration or the effect of a displacement of the grating location in the core with respect to the geometrical center of the fiber.

Fig. 5-30 Experimental FBG sensitivity to small diametric loading force and simulated sensitivity by finite elements for two sets of borosilicate material parameters