## 2 Lateral Stress Sensing Principle  
The spectral peak reflected from the grating is at the Bragg wavelength (\(\uplambda_{\text{B}}\)), which is given by:  
\[\uplambda_{\text{B}}=2\upmu_{\text{eff}}\uplambda \tag{1}\]  
where n\({}_{\text{eff}}\) is the effective refractive index of the core, and \(\uplambda\) is the grating period. According to Equation (1), external factors such as axial strain and temperature variations applied to the FBG can alter the grating period, which would cause the Bragg wavelength to drift.  
Since the grating length is much larger than the cladding diameter, the deformation caused by the lateral load on the fiber can be considered planar deformation. Therefore, the stress in each direction of the FBG can be expressed as:  
\[\left\{\begin{array}{l}\sigma_{\text{x}}(\text{x},\text{y})=\frac{2\text{F }}{\pi\text{hD}}\\ \sigma_{\text{y}}(\text{x},\text{y})=-\frac{6\text{F}}{\pi\text{hD}}\\ \sigma_{\text{z}}(\text{x},\text{y})=\text{v}(\sigma_{\text{x}}+\sigma_{\text {y}})\end{array}\right. \tag{2}\]  
The relationship between refractive index change and stress can be expressed as:  
\[\left[\begin{array}{l}\Delta\upmu_{\text{eff}}\\ \Delta\upmu_{\text{eff}}\end{array}\right]=-\frac{\uplambda_{\text{eff}}^{3 }}{2\text{E}}\left[\begin{array}{ll}\text{P}_{11}-2\text{v}\text{P}_{12}&( 1-\text{v})\text{P}_{12}-\text{v}\text{P}_{11}&(1-\text{v})\text{P}_{12}- \text{v}\text{P}_{11}\\ (1-\text{v})\text{P}_{12}-\text{v}\text{P}_{11}&\text{P}_{11}-2\text{v}\text{P }_{12}&(1-\text{v})\text{P}_{12}-\text{v}\text{P}_{11}\end{array}\right] \left[\begin{array}{l}\sigma_{\text{x}}\\ \sigma_{\text{y}}\\ \sigma_{\text{z}}\end{array}\right] \tag{3}\]  
where E is the Young's modulus of the fiber, v is Poisson's ratio, \(P_{ij}\) is the strain-optic coefficient of the elastic-optic tensor, F is the lateral pressure of the FBG, h is the force length, D is the diameter of the fiber, and \(\sigma_{\text{x}}\), \(\sigma_{\text{y}}\), \(\sigma_{\text{z}}\) are the stresses in x, y, z directions, respectively. For ordinary single-mode fiber, the values for each parameter are as follows: E = 74.52 GPa, \(\text{v}\) = 0.17, P\({}_{11}\) = 0.121, P\({}_{12}\) = 0.270, n\({}_{\text{eff}}\) = 1.447.  
Figure 1 shows the relationship between change in refractive index and force when the FBG is subjected to lateral stress with a force length of 4 mm. The solid black and red lines are the refractive index curves in the x and y polarization directions, respectively. It can be seen that when the FBG is subjected to a linear distribution of external forces, there is a good linear relationship between the change in refractive index and the force. The change  
in refractive index in the x polarization direction is higher than the change in refractive index in the y direction, which is due to the photoelastic coefficient of the fiber \(\mathrm{P_{11}}<\mathrm{P_{12}}\).  
According to Equations (1) and (3), the drift formula for the Bragg reflected wavelength is, in the x direction:  
\[\Delta\lambda_{\mathrm{x}}=-\frac{\mathrm{n_{eff}^{3}}\Lambda}{\mathrm{E}}\{( \mathrm{P_{11}}-2\mathrm{v}\mathrm{P_{12}})\sigma_{\mathrm{x}}+[(1-\mathrm{v} )\mathrm{P_{12}}-\mathrm{v}\mathrm{P_{11}}](\sigma_{\mathrm{y}}+\sigma_{ \mathrm{z}})\} \tag{4}\]  
and in the y direction:  
\[\Delta\lambda_{\mathrm{y}}=-\frac{\mathrm{n_{eff}^{3}}\Lambda}{\mathrm{E}}\{( \mathrm{P_{11}}-2\mathrm{v}\mathrm{P_{12}})\sigma_{\mathrm{y}}+[(1-\mathrm{v} )\mathrm{P_{12}}-\mathrm{v}\mathrm{P_{11}}](\sigma_{\mathrm{x}}+\sigma_{ \mathrm{z}})\} \tag{5}\]  
The lateral stress on the FBG is shown in Figure 2; z is the axial direction of the fiber, y is the direction of the radial force, and x is perpendicular to y. When a certain section of FBG with length h is subjected to a uniform lateral pressure F, the FBG can be approximately decomposed into three sub-grating series, where h\({}_{2}\) is the grating force length, and h\({}_{1}\) and h\({}_{3}\) are the grating unstressed length.