## 2 Principle and Model  
The poling principle of the model is summarized below. At the beginning of poling, in the high temperature and high voltage condition, sodium (Na+) of high mobility is first activated. Driven by the external electric field, sodium (Na+) migrates from the anode to the cathode and the outer surface of the fiber. At this time, cation-depleted region is formed on the outer surface of the anode. As thermal poling proceeds, the water molecules in the air at the surface of the anode ionize into H\({}_{3}\)O+ under the effect of the external electric field, and then accumulate into the fiber. H\({}_{3}\)O+ of lower mobility also begins to migrate toward the cathode and the outer surface of the fiber, gradually neutralizing the negatively charged holes [11] left by the migration of Na+. This promotes the process of the depletion region covering the core. When the depletion region covers the fiber core center, the internal electric field reaches a maximum. The maximum electric field is recorded after the bias is switched off and temperature is lowered to room temperature [12]. The optical fiber is initially neutral and maintains this neutrality all along the poling process [13]. The model is based on the COMSOL finite element method of the transport of diluted species problem, assuming that low concentration ions (-1 ppm) move in the presence of diffusion and drift due to an electric field [8]. Both migration and diffusion are taken into account, which can be derived by coupling the local equation of continuity to [PERSON]'s equation [14]:  
\[ \begin{array}{c}\frac{\partial\text{C}_{\text{i}}}{\partial\text{t}}+\
abla \cdot(-\text{D}_{\text{i}}\
abla\text{C}_{\text{i}}-\text{z}_{\text{i}}\mu_{ \text{i}}\text{FC}_{\text{i}}\
abla\Phi)=\text{R}_{\text{i}}\\ \
abla^{2}\Phi=-\frac{\text{F}}{\epsilon}\sum\limits_{\text{i}}\text{z}_{\text {i}}\text{C}_{\text{i}}\end{array}  \tag{1}\]  
where the index i represents different ionic species, C\({}_{\text{i}}\) is ionic concentration, D\({}_{\text{i}}\) is the diffusivity, z\({}_{\text{i}}\) is the charge, \(\mu_{\text{i}}\) is the ionic mobility, F is [PERSON]'s constant, \(\Phi\) is the electric potential during poling, and R\({}_{\text{i}}\) is the production or consumption rate. D\({}_{\text{i}}=\text{k}_{\text{B}}\tau\mu_{\text{i}}/\text{e}\), k\({}_{\text{B}}\) is the Boltzmann constant, T is the temperature. The first term in the local equation of continuity explains the transient of the ionic, and the second term explains the drift and diffusion of the charge. The [PERSON]'s equation shows that the charge distribution determines the electric field distribution. Combined with the equation \(\chi^{(2)}_{\text{eff}}=3\chi^{(3)}\cdot\text{E}_{\text{rec}}\), the evolution of second-order nonlinearity of optical fiber by thermal poling can be obtained [15]. The material type used in the simulation for the fibers is silica glass with a refractive index of 1.468. The two models use three modules: electrostatics, transport of diluted species and general form boundary partial differential equation. The electrostatics module mainly sets the electric potential and the space charge density of the fiber, wherein the space charge density changes with the concentration of Na+ and H\({}_{3}\)O+. The transport of diluted species module mainly sets the concentration, diffusion coefficient and mobility of Na+ and H\({}_{3}\)O+ in the fiber. The general form boundary PDE module sets the concentration of H\({}_{3}\)O+ at the holes boundary \(\frac{\partial\text{C}_{\text{i}}}{\partial\text{t}}=\sigma_{2}\)E [13]. The poling conditions of the two models are the same: assuming that the holes are filled with metal electrodes [16], the poling temperature is 285 \({}^{\circ}\)C, the outer surface of the fiber is set to zero potential, and the potential of the two electrodes are uniformly 5 KV.