## 1 Introduction  
Research of integrated optics has developed rapidly [1, 2, 3]. The diverse passive and active nanophotonic devices based on silicon-on-insulator (SOI) waveguides and silicon nitride (Si3N4) waveguides such as modulators [4, 5], filters [6, 7], wavelength division multiplexers [8, 9], and detectors [10, 11] have been widely developed and have shown numerous applications in many fields. However, challenges still exist. Loss of the large-scale on-chip integrated devices limits the device performance and increases the bit error rate of the signal, which hinders the development of large-scale integration [12, 13]. On-chip erbium-doped waveguide optical amplifier (EDWA) can compensate for the loss effectively [14, 15, 16, 17]. It is an efficient way to overcome this problem. Si and Si3N4 cannot efficiently perform the task of integrating optical interconnections alone. Therefore, it is highly necessary to integrate with other active materials. The novel slot waveguide, which can strongly confine the light in a narrow slot region of low refractive index material [18, 19, 20], is more likely to integrate with other materials than the traditional strip waveguide. The density of the light field in the slot is nearly 20 times higher than that of a conventional rectangular dielectric waveguide and will promote the signal and pump interaction with the gain medium. This kind of waveguide amplifier based on slots is of great value for improving the gain performance and reducing the threshold pump power. In recent years, few reports regarding the optical waveguide amplifiers based on slots integrated with erbium-doped materials have been reported. In 2012, the Er/Yb silicate slot waveguide amplifier was fabricated and achieved a signal 1.7 dB gain at 1.53 um in a waveguide with a 6 mm long slot pumped at 1476 nm when the Er3+ concentration was confirmed to be \(1\times 10^{27}\) m\({}^{-3}\)[21]. However, the inorganic material is difficult to integrate with slotsbecause of the complicated technology, expensive equipment, and long cycle. Erbium-doped polymer materials have the outstanding advantages of high bandwidth, low cost, species diversity, and ease of realizing large-scale and high-density integration [22; 23; 24; 25]. Integrating with slots for EDWA fabrication is a good decision. In 2014, the potential for large optical gain improvement of erbium-doped rich polymers integrated into silicon-slot waveguides amplifier pumped at 1480 nm was shown [26]. In 2020, the gain characteristics of the hybrid slot waveguide amplifiers integrated with NaYF\({}_{4}\):Er\({}^{3}\)+ NPs-PMMA pumped at 1480 nm were analyzed. A net gain of 5.78 dB was achieved with the length of 1.5 cm when the Er\({}^{3+}\) concentration was 1.3 \(\times\) 10\({}^{27}\) m\({}^{-3}\)[27]. In this case, the gain of the slot waveguide amplifier with the polymer as gain medium is still not ideal. This is mainly because Si has strong absorption of 980 nm light. Therefore, 1480 nm was chosen as a pump and pure erbium-doped polymer as gain medium. The low-pumping efficiency at 1480 nm and the low-Er\({}^{3+}\)-luminescence intensity of pure erbium-doped materials without Yb\({}^{3+}\) sensitization limited the amplification properties which could not meet the requirements of optical communications. To overcome this issue, a Si\({}_{3}\)N\({}_{4}\) platform with low loss and a weak absorption effect on 980 nm was employed [28; 29]. A Si\({}_{3}\)N\({}_{4}\) slot instead of slot-based Si confined the light field in the slot region tightly. Er\({}^{3+}\) and Yb\({}^{3+}\)codoped polymer material was used as gain medium. In this work, the core-shell \(\alpha\)-NaYF\({}_{4}\)/ \(\beta\)-NaLuF\({}_{4}\): Yb\({}^{3+}\), Er\({}^{3+}\) nanocrystals were synthesized using a mild hydrothermal method and then copolymerized with methyl methacrylate (MMA) to fabricate NaYF\({}_{4}\)/NaLuF\({}_{4}\): Yb, Er NPs-PMMA as gain medium.