## 2 FBG Sensing Principle  
The peak wavelength of optical reflection (Bragg wavelength) in FBG depends on the grating period of refractive index and core refractive index, which will be directly determined by temperature and strain. Accordingly, temperature and strain are two basic factors that will cause the change in the FBG Bragg-wavelength response and should be considered for FBG sensing measurements of other physical quantities. The strain-caused variation in the Bragg wavelength of FBG sensor for current detection originates from the elasto-optical effect of fiber stretching (grating period change) and core refractive index. When the fiber is subjected to axial (set as z coordinate) stress, the produced lateral stretching or compressing strain results in the variation of the FBG grating period, while the change in the effective refractive index in the fiber core \(n_{\text{eff}}\) can be deduced from the elasto-optical tensor and the strain tensor as follows:  
\[\Delta\bigg{(}\frac{1}{n_{\text{eff}}}\bigg{)}_{i}^{2}=\sum_{j=1}^{6}P_{ji} \varepsilon_{ij}\quad(i=1\to 6) \tag{1}\]  
where the subscripts \(I\) = 1, 2, 3 and 4, 5, 6 indicate, respectively, the real and imaginary (the electric field attenuation of incident light) parts of the complex refractive index along three directions in the fiber core. There is no shear stress when the fiber is stretched, and the strain-caused change in the refractive index along the fiber axial direction for isotropic material can be predigested as:  
\[\Delta n_{\text{eff}}=-\frac{1}{2}n_{\text{eff}}^{3}\Delta\bigg{(}\frac{1}{ \Delta n_{\text{eff}}}\bigg{)}_{\text{x,y}}=-\frac{1}{2}n_{\text{eff}}^{3}[P_{ 12}-\
u(P_{11}+P_{12})]\varepsilon_{\text{z}} \tag{2}\]  
where \(P_{11}\) and \(P_{12}\) denote three no-zero elasto-optical coefficients, \(\
u\) symbolizes Poisson's ratio of the fiber core, and \(\varepsilon_{\text{z}}\) represents the strain along the fiber axis. Accordingly, the effective elasto-optical coefficient \(P_{\text{eff}}\) can be defined by the following equation:  
\[P_{\text{eff}}=\frac{n_{\text{eff}}^{2}}{2}[P_{12}-\
u(P_{11}+P_{12})] \tag{3}\]  
The strain-caused variation in Bragg wavelength of FBG reflection is finally presented by defining the strain sensitivity in the first and second order \(K_{\text{e}}{{}^{\prime}}\) and \(K_{\text{e}}{{}^{\prime\prime}}\), respectively, as follows:  
\[\begin{array}{l}\Delta\lambda_{\text{B}}=(K_{\text{e}}{{}^{\prime}}\varepsilon _{\text{z}}+\frac{1}{2}K_{\text{e}}{{}^{\prime\prime}}\varepsilon_{\text{z}} {{}^{2}})\lambda_{\text{B}}\\ K_{\text{e}}{{}^{\prime}}=\frac{\Delta\lambda_{\text{B}}}{\lambda_{\text{B}} \varepsilon_{\text{z}}^{2}}=1-P_{\text{eff}}\\ K_{\text{e}}{{}^{\prime\prime}}=\frac{\Delta^{2}\lambda_{\text{B}}}{\lambda_{ \text{B}}\varepsilon_{\text{z}}^{2}}=(1-P_{\text{eff}})^{2}+2P_{\text{eff}}{{} ^{2}}\end{array} \tag{4}\]  
where \(\lambda_{\text{B}}\) and \(\Delta\lambda_{\text{B}}\) indicate the Bragg wavelength and its variation, respectively. For the germanium-doped quartz of the core material in a representative optical fiber, \(P_{\text{eff}}\) = 0.22 and the first-order strain sensitivity \(K_{\text{e}}{{}^{\prime}}\) = 0.78, while the second-order sensitivity \(K_{\text{e}}{{}^{\prime\prime}}\) is so small that it can be ignored in practical applications, by which the Bragg wavelength can be considered as being changed linearly with fiber axial strain. To this end, an FBG Bragg reflection wavelength is determined by an effective refractive index and grating period of a single-mode fiber, as described by Equations (1)-(4). As the FBG is subjected to stress induced by the target, detecting physical quantities along the fiber axial direction at a certain ambient temperature, the effective refractive index and grating period will accordingly change to shift the Bragg wavelength and realize the strain-wavelength modulations.