## 2 Experiments Details
### Optimization of the Slot Waveguide
#### 2.3.1 Optimization of Slot Waveguides Combined with Maximum Confinement Factors  
For a more intuitive view of the slot's ability to restrict the light, we defined the overlapping integration factor \(\Gamma_{\mathrm{S}}\) of the slot waveguide to represent the effective utilization of energy, as shown in Equation (1), so we searched for a maximum in all the considered cases, which could be used to achieve higher gain [30]. The finite difference method (FDM) was used to calculate \(\Gamma_{\mathrm{S}}\) to primarily optimize the height of Si\({}_{3}\)N\({}_{4}\) (H\({}_{\mathrm{r}}\)), the width of Si\({}_{3}\)N\({}_{4}\) (W\({}_{\mathrm{r}}\)), and the slit width (W\({}_{\mathrm{S}}\)). \[\Gamma_{S}=\frac{\iint_{\mathrm{S}}\left|E(x,y)\right|^{2}dxdy}{\iint_{Total} \left|E(x,y)\right|^{2}dxdy} \tag{1}\]  
First, the effective refractive index (N\({}_{\mathrm{eff}}\)) and \(\Gamma_{\mathrm{S}}\) were calculated as a function of the width of W\({}_{\mathrm{r}}\) at 1530 nm and 980 nm under TE and TM polarizations shown in Figure 7a,b. The wavelength interval between the 980 nm pump light and signal light was quite large; the optical field distribution was significantly different. Then, the N\({}_{\mathrm{eff}}\) and \(\Gamma_{\mathrm{S}}\) of the waveguide at signal wavelengths were quite different from those of pump wavelengths. Under the TE polarization mode, the light field was confined in the slot well. With increasing W\({}_{\mathrm{r}}\), Ne\({}_{\mathrm{eff}}\) also increased, and \(\Gamma_{\mathrm{S}}\) at 980 nm gradually decreased. At the same time, \(\Gamma\)S at 1530 nm tended to saturate when W\({}_{\mathrm{r}}\) increased to 400 nm. This is due to the low impact of W\({}_{\mathrm{r}}\) at 1530 nm on the waveguide light field. At 980 nm, the light field was mostly confined to the Si\({}_{3}\)N\({}_{4}\) strip waveguide. As Wr increased, more light fields were confined to the Si\({}_{3}\)N\({}_{4}\) strip waveguide. As a result, N\({}_{\mathrm{eff}}\) increased while \(\Gamma_{\mathrm{S}}\) gradually decreased. Next, we optimized WS combined with the N\({}_{\mathrm{eff}}\) and \(\Gamma_{\mathrm{S}}\) curves at the signal and pump wavelengths under TE and TM polarizations, as revealed in Figure 7c,d. The curves under TE polarization tend to be basically consistent with that of TM polarization. When W\({}_{\mathrm{S}}\) increased, N\({}_{\mathrm{eff}}\) decreased. Additionally, \(\Gamma_{\mathrm{S}}\) at a 980 nm wavelength decreased rapidly, increasing W\({}_{\mathrm{S}}\). It increased to extreme values and then decreased rapidly. \(\Gamma_{\mathrm{S}}\) increased rapidly at 1530 nm as W\({}_{\mathrm{S}}\) increased and tended to be saturated. Maximum \(\Gamma_{\mathrm{S}}\) was obtained when W\({}_{\mathrm{S}}\) was 200 nm. This is because with increased W\({}_{\mathrm{S}}\), light field leakage to the upper and lower cladding was gradually controlled, and more light fields were included in the slot. Therefore, \(\Gamma_{\mathrm{S}}\) increases rapidly at this time, accompanied by a rapid decrease in N\({}_{\mathrm{eff}}\). However, with increased W\({}_{\mathrm{S}}\), the coupling resonance of the abrupt light field at the interface between slot dielectrics was weakened. Thus, the confinement of the light field in the slot was impaired, and some light fields leaked to the cladding. As a result, N\({}_{\mathrm{eff}}\) and \(\Gamma_{\mathrm{S}}\) gradually decreased, and the rate of decrease in N\({}_{\mathrm{eff}}\) became slower. Combined with the dependence of N\({}_{\mathrm{eff}}\) and \(\Gamma_{\mathrm{S}}\) on the different widths of H\({}_{\mathrm{r}}\) shown in Figure 7e,f, we optimized the H\({}_{\mathrm{r}}\) of the waveguide. H\({}_{\mathrm{r}}\) is the main factor affecting \(\Gamma_{\mathrm{S}}\). For any waveguide slot size, increasing its H\({}_{\mathrm{r}}\) will make \(\Gamma_{\mathrm{S}}\) increase. As H\({}_{\mathrm{r}}\) increased, both N\({}_{\mathrm{eff}}\) and \(\Gamma_{\mathrm{S}}\) increased monotonically.