## 3 Results  
The novel Er and Yb codoped waveguide amplifiers were presented by integrating NaYF\({}_{4}\)/NaLuF\({}_{4}\): Er\({}^{3+}\), Yb\({}^{3+}\) NCs-PMMA with a Si\({}_{3}\)N\({}_{4}\) slot waveguide. To analyze the gain performance of the device accurately, we established a transition model of an Er-Yb codoped six-level system pumped at 980 nm, as shown in Figure 9a. The Yb\({}^{3+}\) ions in the ground state \({}^{2}\)F\({}_{7/2}\) were excited by a 980 nm pump absorbing the energy and then transitioned upward to level \({}^{2}\)F\({}_{5/2}\). Because \({}^{2}\)F\({}_{5/2}\) and \({}^{2}\)F\({}_{7/2}\) levels of Yb\({}^{3+}\) had the same energy spacing between Er\({}^{3+}\)\({}^{4}\)I\({}_{11/2}\) and \({}^{4}\)I\({}_{15/2}\) levels, and erbium yterbium atoms were very far apart, Yb\({}^{3+}\) ions rapidly transmitted energy to Er\({}^{3+}\) ions in the ground state by cross-relaxation. Thus, they transited from the \({}^{4}\)I\({}_{15/2}\) level to the excited-state level \({}^{4}\)I\({}_{11/2}\). Due to the instability of \({}^{4}\)I\({}_{15/2}\), Er\({}^{3+}\) ions quickly transferred to the metastable state \({}^{4}\)I\({}_{13/2}\) through the nonradiative relaxation to form population inversion. Then, they transitioned down to the ground state \({}^{4}\)I\({}_{15/2}\) by excited emission and emitted photons with the same frequency as the signal light. The amplification of the signal light was realized. To build an accurate simulation model, we also considered cooperative up-conversion C\({}_{\mathrm{up}}\) between metastable levels \({}^{4}\)I\({}_{13/2}\) and \({}^{4}\)I\({}_{9/2}\), \({}^{4}\)I\({}_{15/2}\). Spontaneous radiation transitions from \({}^{4}\)I\({}_{9/2}\), \({}^{4}\)I\({}_{11/2}\), and \({}^{4}\)I\({}_{13/2}\) to their next levels were taken into account, as well as \({}^{2}\)F\({}_{5/2}\)Yb to \({}^{2}\)F\({}_{7/2}\)Yb. Amplified spontaneous emission (ASE) was neglected. A six-level spectroscopic model pumped at 980 nm was presented. The rate equations and propagation equations were solved. Incorporated with the characteristic parameters of the gain medium, the gain G at any point of the amplifier in the transmission direction z was calculated by the fourth-order Runge-Kutta method. G can be expressed in dB as:  
\[G=10\lg\frac{P_{S}(z)}{P_{S}(0)} \tag{3}\]  
where \(P_{S}(0)\) is the input signal power and \(P_{S}(z)\) is the output signal power. As illustrated in Equation (3), a higher gain can be achieved with small input signal power. In combination with experimental experience, a 0.1 mW input power was selected. The Si\({}_{3}\)N\({}_{4}\) slot waveguide amplifier was fabricated via the following steps: first, the Si\({}_{3}\)N\({}_{4}\) film on SiO\({}_{2}\) was first annealed at a high temperature of 1050-1150 \({}^{\circ}\)C for 3-7 h to remove the hydrogen bonds on the surface and inhibit the formation of O-H and N-H bonds before preparing the waveguide. The positive electron resistance ZEP520 with a thickness of 400 nm was spin-coated on the Si\({}_{3}\)N\({}_{4}\) wafer. Then, the standard pattern exposure was performed using an electron beam. After image development and ICP (SPTS) etching for 140 s, the Si\({}_{3}\)N\({}_{4}\) slot waveguide was obtained. Finally, a 3 \(\upmu\)m-thick PMMA film was spin-coated onto the Si\({}_{3}\)N\({}_{4}\) slot as the core layer and then baked at 120 \({}^{\circ}\)C for 2 h. The Si\({}_{3}\)N\({}_{4}\) slot waveguide amplifier was obtained. We simulated the gain versus coordinate waveguide length with different transmission losses in Figure 9b. The optimal waveguide length corresponding to different transmission losses was different. The corresponding optimal waveguide length was longer when the transmission loss was smaller. When the transmission loss was only 2 dB/cm, the optimal waveguide length was approximately 8 cm, and a net gain of nearly 20 dB could be obtained. As transmission losses gradually increased, the gain values gradually decreased. No net gain was generated when the transmission loss was 5 dB/cm.