## 4 Multi-Physics Simulation Analyses of GMM-FBG Current Sensor
### Temperature Drift Characteristics of GMM-FBG Bragg Wavelength  
In the modeled GMM-FBG optical modulator, the FBG is directly contacted with the GMM rod on the side surfaces to construct a GMM-FBG strain-optical modulation component capable of effectively transferring magnetostrictive strain along the axial direction in the GMM rod under current excitation, as shown in Figure 5. The rectangular groove in a dimension of 0.2 mm \(\times\) 0.2 mm \(\times\) 13 mm is excavated out on the side surface and along the longitudinal axis of the GMM rod, in which a bare FBG (without an outer protective layer) with a tail fiber is placed and fixed by a filling binder in the interval space. The FBG in a sufficient length of 10 mm is located at the center of the long and narrow groove with two residual spaces of 1.5 mm length at the FBG terminals to accommodate the tail fibers, as shown in Figure 5. Epoxy resin material is used as a binder for fixing and protecting the bare FBG. The FBG is specified by the standard 9/125 \(\upmu\)m single-mode fiber: the core diameter and cladding thickness are 9 \(\upmu\)m and 125 \(\upmu\)m, respectively; the core material is defined by germanium-doped quartz as an elasto-optical coefficient of \(P_{11}\) = 0.121 and \(P_{12}\) = 0.27, Poisson's ratio of \(\
u\) = 017, and refractive index of \(n\) = 1.469 (1550 nm). For specifying boundary conditions in finite-element simulations, the solid mechanical model of GMM-FBG fixation is constructed by setting the joint constraint (relative fixed constraint) to the contacting interfaces of the FBG cladding layer/solid binder, GMM/binder (five faces of groove), and FBG core/cladding. Due to the difference in the thermal expansion coefficients of FBG and GMM, the refractive index of the FBG core will be changed with the elasto-optical effect when thermal stress arises at the interface between FBG and GMM from temperature variation, while the Bragg wavelength temperature drift deriving from magnetostriction and thermal expansion in the ferrite-GMM assemblage is attributed to the temperature dependence of the current-magnetic-strain sensing response and static working point, which can be directly evaluated by equation (a) according to the simulation results in Section 4.3. Therefore, due to a particular interest of investigating the temperature drift caused by thermal stress in the FBG  
Figure 5: GMM-FBG coupled configuration of ferrite–GMM-FBG current sensor model. GMM optical modulator, the multi-physics simulations of magnetostriction and thermal stress coupled fields in a ferrite-GMM-FBG current sensor are performed without modeling the thermal expansion in a ferrite magnetizer. According to the elasto-optical effect, the refractive index of the FBG core being altered by thermal stress in the GMM-FBG optical modulator can be calculated from the elasto-optical tensor of the FBG core material with Equation (2) and the results of the thermal stress/strain simulations with material characteristics specified for the GMM-FBG modulator as listed in Table 2. The small thermal expansion coefficients of silica quartz glasses making up FBG imply that no considerable expansion or shrinkage will appear when the temperature changes. In contrast, the thermal expansion coefficient of epoxy resin used for gluing FBG on the GMM side in the groove is significantly higher than that of FBG. As a result, the thermal stress/strain along the longitudinal axis will be produced under the restraint of the GMM rod when the temperature changes. The sensing characteristics of the GMM-FBG current sensor represented by the Bragg wavelength as a function of the excitation current at various temperatures are shown in Figure 6. The Bragg wavelength differential signal sensitive to static current represents a substantial abatement by temperature elevation in both the gradient and amplitude of response curves, leading to the foreseeable reductions of sensitivity and range, which means the GMM-FBG current sensor needs to be calibrated according to the work temperature.