# A Strain-Transfer Model of Surface-Bonded Sapphire-Derived Fiber Bragg Grating Sensors
## 4 Experiment and Discussion  
As listed in Table 3, three different kinds of SDFBGs with the same cladding radius were used in uniaxial tensile tests. Among them, SDBG-1 had the highest alumina content and largest core radius. Compared to SDFBG-1, SDFBG-2 had similar alumina content but a smaller core radius, while SDFBG-3 had lower alumina content but a similar core radius. These gratings were all fabricated with a 1550 nm central wavelength and 10 mm length by a near-infrared femtosecond laser in accordance with the point-by-point method. Strain measured by the SDFBG sensor is expressed as  
\[\varepsilon=\frac{\Delta\lambda}{K}, \tag{18}\]  
where \(\Delta\lambda\) is the resonant-wavelength shift of the SDFBG induced by axial strain, and \(K\) is the axial-strain sensitivity, which is defined by [32]  
\[K=\frac{1}{\lambda_{0}}\Big{(}1-\frac{1}{2}n\Big{(}p_{12}^{core}-v^{core}\Big{(} p_{11}^{core}+p_{12}^{core}\Big{)}\Big{)}\Big{)}, \tag{19}\]  
where \(\lambda_{0}\) is the selected resonant wavelength. Refractive index \(n\), two photoelastic constants \(p_{12}^{core}\) and \(p_{11}^{core}\) and Poisson ratio \(v^{core}\) of the core were all determined by alumina content \(m\) and could be calculated by using the additive model [30] via each component's bulk parameter in Table 4:  
\[\left\{ \begin{array}{l}n=mn_{Al_{2}O_{3}}+(1-m)n_{SiO_{2}}\\ p_{11}^{core}=mp_{11}^{Al_{2}O_{3}}+(1-m)p_{11}^{SiO_{2}}\\ p_{12}^{core}=mp_{12}^{Al_{2}O_{3}}+(1-m)p_{12}^{SiO_{2}}\\ v^{core}=mv^{Al_{2}O_{3}}+(1-m)v^{SiO_{2}}\end{array} \right. \tag{20}\]  
The SDFBG was mounted onto a GH3536 specimen in load control to calibrate strain sensitivity using a universal testing machine C40 (MTS Systems, Co., Minnesota, USA). The schematic diagram and experiment setup are shown in Figures 8 and 9, respectively. The adhesive was fabricated with half-bond length of 5 mm and bottom thickness of 0.1 mm. Static-displacement load was applied to the upper end of the specimen, while the tether end was fixed. Loaded strain was defined as \(\varepsilon=\Delta L_{m}/L_{m}\), where \(L_{m}\) is the length of the specimen between two clamps at the initial stage, and \(\Delta L_{m}\) is the elongation of the specimen under static tensile loading. The reflection spectrum of the SDFBG was adopted by an optical sensing analyzer Si720 (Micron Optics, Inc., Atlanta, USA) with a spectral resolution of 0.25 pm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(n\) & \(\
u\) & \(p_{11}\) & \(p_{12}\) \\ \hline Al\({}_{2}\)O\({}_{3}\) & 1.667 & 0.25 & \(-\)0.237 & \(-\)0.027 \\ SiO\({}_{2}\) & 1.444 & 0.16 & 0.098 & 0.226 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Bulk parameters of alumina and silica. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Alumina Content (mol \%)** & **Core Radius (\(\mu\)m)** & **Cladding Radius (\(\mu\)m)** \\ \hline SDFBG-1 & 49.6 & 30.8 & 62.5 \\ SDFBG-2 & 49.3 & 10.5 & 62.5 \\ SDFBG-3 & 4.5 & 30.2 & 62.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Basic parameters of sapphire-derived fiber Bragg gratings (SDFBGs). The strain experiment was carried out at room temperature (about 20 \({}^{\circ}\)C) from 0 to 1000 \(\mu\varepsilon\), with a step of 100 \(\mu\varepsilon\). Theoretical and experiment results are listed in Table 5. Experimental ASTRs were obtained by dividing theoretical sensitivities by the corresponding experimental sensitivities. Compared to SDFBG-2, SDFBG-1 had a 20.3 \(\upmu\)m larger core radius, but 5.4% lower experimental ASTR. Meanwhile, compared to SDFBG-3, SDFBG-1 had a 45.1 mol % higher alumina content, but a 3.6% lower experimental ASTR. As expected, the larger core diameter and higher alumina content led to lower ASTRs.