# Femtosecond-Laser-Assisted Fabrication of Radiation-Resistant Fiber Bragg Grating Sensors
## 2 Fundamental Concepts  
When a broadband light source is transmitted through a FBG-inscribed optical fiber, a particular wavelength, known as the Bragg wavelength (\(\lambda_{B}\)), of the incident broadband light source obeys the following phase matching condition [1]. \[\lambda_{B}=2n_{eff}\Lambda \tag{1}\]  
where \(n_{eff}\) represents the effective refractive index of the FBG region of the fiber core and \(\Lambda\) represents the gratings' period. The Bragg wavelength undergoes strong reflection from the FBG while the remaining spectrum transmits through the FBG. The Bragg wavelength can be shifted by controlling the gratings' period and/or the effective refractive index of the FBG. Temperature sensitivity is inherent to fiber Bragg gratings where the primary contributor to Bragg wavelength shift is the variation in the \(n_{eff}\) in the FBG region caused by the mechanical stress (\(\sigma\)) and temperature (\(T\)). In addition, thermal expansion alters the gratings' period, which is less significant due to the low thermal expansion coefficient of silica fiber. A change in the refractive index (\(\Delta n\)) can be estimated by the following Equation [2]. \[\Delta n(\Delta\sigma,\Delta T)=n(\sigma,T)-(\sigma_{ref},T_{ref})=\left[ \frac{\partial n}{\partial\sigma}\right]_{T}\Delta\sigma+\left[\frac{\partial n }{\partial T}\right]_{\sigma}\Delta T \tag{2}\]  
where \(\sigma_{ref}\) and \(T_{ref}\) refer to the reference values of the local stress and temperature, respectively, and \(\Delta\sigma\) and \(\Delta T\) represent the incremental deviations in the local stress and temperature from their reference values, respectively. If the change in the Bragg wavelength is denoted by \(\Delta\lambda_{B}\), Equation (2) can be elaborated as follows [2]:  
\[\frac{\Delta\lambda_{B}}{\lambda_{B}}=\left[\left[\frac{\varepsilon}{\partial \sigma}\right]_{T}+\frac{1}{n_{eff}}\left(\frac{\partial n_{eff}}{\varepsilon }\right)_{T}\left(\frac{\varepsilon}{\partial\sigma}\right)_{T}\right]\Delta \sigma+\left[\left(\frac{\varepsilon}{\partial T}\right)+\frac{1}{n_{eff}} \left(\frac{\partial n_{eff}}{\partial T}\right)_{\sigma}\right]\Delta T \tag{3}\]where \(\epsilon=\pm\partial\Lambda/\Lambda\) is the strain with the \(\pm\)sign exemplifying the tensile and compressive stresses. Equation (3) can be rewritten using [PERSON]'s modulus (\(Y\)) and the thermal expansion coefficient (\(\alpha\)) [2,3]. \[ \begin{array}{rl}\frac{\Delta\lambda_{B}}{\lambda_{B}}&=\frac{1}{Y}\Big{[}1+ \frac{1}{n_{eff}}\Big{(}\frac{\partial n_{eff}}{\epsilon}\Big{)}_{T}\Big{]} \Delta\sigma+\Big{[}\alpha+\frac{1}{n_{eff}}\Big{(}\frac{\partial n_{eff}}{ \partial T}\Big{)}_{\sigma}\Big{]}\Delta T\\ &=\epsilon_{z}-\frac{n_{eff}^{2}}{2}\big{[}\epsilon_{r}(P_{11}+P_{12})+P_{12} \epsilon_{z}\big{]}+[\alpha+\zeta]\Delta T\\ &=(1-P_{e})\epsilon_{z}+(\alpha+\zeta)\Delta T\end{array}  \tag{4}\]  
where \(\epsilon_{z}\) is the axial strain, \(\epsilon_{r}\) is the radial strain, \(P_{11}\) and and \(P_{12}\) are [PERSON]'s coefficients, \(\alpha=0.55\times 10^{-6}/^{\circ}\)C (for silica fiber) is the thermal expansion coefficient, \(\zeta=5.77\times 10^{-6}/^{\circ}\)C (for silica fiber) is the thermo-optic effect, and \(P_{e}\) is the photo-elastic coefficient of the fiber (\(P_{e}\approx 0.22\) for Ge-doped silica fiber). Thus, from Equation (4), the temperature dependence of FBG sensors, i.e., the Bragg wavelength shift due to a change in the temperature, can be determined by the following equation. \[\frac{\Delta\lambda_{B}}{\lambda_{B}}=(\alpha+\zeta)\Delta T \tag{5}\]  
Figure 1a illustrates the basic principle of FBG sensors for measuring temperature sensitivity. We utilized the phase mask technique [30,34,35,36] to fabricate the gratings inside the fused silica glass sample and optical fibers. The \(+1/-1\)-order phase mask was optimized to diffract light equally into the \(+1\) and \(-1\) orders where the 0-order beam was highly suppressed.