# Optical Fiber Sensor with Stable Operating Point for AC Magnetic Field Measurement
## 2 Theoretical Analysis  
The proposed sensor is based on a fiber Bragg grating (FBG) fabricated on the core of a high attenuation fiber (HAF) that is bonded onto a magnetostrictive rod surface. The HAF has a high concentration of doping that can be efficiently heated by absorbing energy from a heating laser, an effect that can be utilized to mitigate effects of shifts in the ambient temperature and thus realize a stable operating point. The heat generated by absorbing power from the heating laser predominately affects the effective refractive index of the HAF core--that is, it affects the resonant Bragg wavelength of FBG. The Bragg wavelength \(\lambda_{B}\), at which light traveling through the HAF Bragg grating (HAFBG) will be reflected, is defined as:  
\[\lambda_{B}=2\cdot n_{\textit{eff}}\cdot\Lambda, \tag{1}\]  
where \(n_{\textit{eff}}\) is the effective refractive index of the core of the HAF and \(\Lambda\) is the period of the refractive index modulation of the HAFBG. When an external magnetic field acts on the sensor, the length of magnetostrictive rod will change due to the magnetostriction effect. The strain of the magnetostrictive rod is related to the magnetic field as [34]:  
\[\varepsilon=\frac{\Delta L}{L}=C_{f}B^{2}, \tag{2}\]  
where \(\varepsilon\) is the strain of magnetostrictive rod, and \(\Delta L\) and \(L\) are the elongation length and original length of the magnetostrictive rod, respectively. \(C_{f}\) is a coefficient related to the magnetostriction effect, which is linearly dependent on the magnetic field intensity \(B\) within a specific magnetic field range. \(B\) can be expressed as:  
\[B=B_{0}+A\cdot\cos(\omega_{B}t), \tag{3}\]  
where \(B_{0}\) is the DC offset, and \(A\) and \(\omega_{B}\) are the amplitude and angular frequency of the applied magnetic field, respectively. When the magnetostrictive rod is strained, the HAFBG to which it is attached will also experience a strain \(\varepsilon\), inducing changes in \(n_{\textit{eff}}\) and \(\Lambda\) that result in a wavelength drift of \(\lambda_{B}\). The wavelength drift of \(\lambda_{B}\) is also influenced by the changes in the ambient temperature and can thus be described as:  
\[\Delta\lambda_{B}=\frac{\partial\lambda_{B}}{\partial\varepsilon}\varepsilon+ \frac{\partial\lambda_{B}}{\partial T}T, \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**References** & **Year** & **Sensing Structure** & **Effective Sensing** & **Sensitivity** & **Temperature Compensation** \\  & & & **Length (mm)** & **(pm/mT)** & **Method** \\ \hline
[12] & 2011 &
\begin{tabular}{c} Etched FBG immersed in \\ magnetic fluid \\ \end{tabular} & No mention & 3.44 & No \\ \hline
[14] & 2009 & \
\begin{tabular}{c} Etched FBG coated with \\ TbDyFe/FeNi thin film \\ \end{tabular} & 15 & 1.08 &
\begin{tabular}{c} A separate FBG \\ temperature sensor \\ \end{tabular} \\ \hline
[33] & 2000 & \
\begin{tabular}{c} Standard FBG bonded onto \\ Terfenol–D alloy surface \\ \end{tabular} & 25 & 8.68 &
\begin{tabular}{c} A separate FBG \\ temperature sensor \\ \end{tabular} \\ \hline
[20] & 2021 & \
\begin{tabular}{c} FBG embedded in \\ Terfenol–D composite \\ \end{tabular} & 16 & 2.70 &
\begin{tabular}{c} A separate FBG \\ temperature sensor \\ \end{tabular} \\ \hline
[30] & 2022 &
\begin{tabular}{c} FBG embedded in \\ Terfenol–D composite \\ \end{tabular} & 40 & 9.83 & No \\ \hline
**This work** & \
\begin{tabular}{c} **HAFBG bonded onto** \\ **Terfenol-D alloy surface** \\ \end{tabular} & **10** & **3.95** &
\begin{tabular}{c} **Photo-thermal** \\ **effect of HAF** \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the representative FBG-based magnetic field sensors. where \(\Delta\lambda_{B}\) is the offset value of the wavelength drift of \(\lambda_{B}\) and \(T\) the ambient temperature.