## 3 Experimental Results  
The performance of the Tm:YAP laser pumped by a diode laser is shown in Figure 2. When the incident pump power was 49.5 W, the maximum continuous-wave (CW) output power of 18.1 W was obtained. At the same time, the optical efficiency and the slope efficiency were 31.8% and 43.4%, respectively, which were calculated by linear fitting of the experimental data with respect to the incident diode laser power. A spectrum analyzer (WA-650, EXFO, Quebec, PQ, Canada) combined with an wave meter (WA-1500, EXFO, Canada) was used to measure the output spectrum of the Tm:YAP laser. The central wavelength of the Tm:YAP laser was 1939.2 nm, which matched the absorption line of the Ho:SSO, as shown in Figure 2. The absorption cross section \(\sigma_{abs}\) is 0.79 \(\times\) 10\({}^{-20}\) cm\({}^{2}\) at the pump wavelength. Figure 1: Experimental setup of an single-longitudinal-mode (SLM) Ho:Sc\({}_{2}\)SiO\({}_{5}\) (Ho:SSO) laser resonantly pumped by a Tm:YAP laser. Figure 2: The performance of the Tm:YAP laser. Inset: The absorption spectrum of the Ho:SSO and the pump spectrum (the blue line is the absorption line around 1949 nm, and the red line is the emission spectrum of the Tm:YAP laser). The laser performance under the multi-mode operation is shown in Figure 3. In the experiment, a power meter (Power Max PS19, Coherent, Portland, OR, USA) was used to measure the output power of the Ho:SSO laser. For the Ho:SSO laser, the output power reached a maximum of 2.98 W with an incident power of 16.2 W. At the same time, the optical-optical conversion efficiency was 18.4%. By linear fitting of the experimental data, the slope efficiency was calculated to be 23%. The spectrum analyzer (Model 721, Bristol Instruments, Victor, NY, USA) was used to record the laser spectrum when the incident pump power was 16.2 W. The central wavelength was 2111 nm with an FWHM of 1.1 nm. The result is shown in Figure 3.  
The operation in the free-running mode was immediately achieved for the Ho:SSO laser developed in this study. A Fabry-Perot interferometer whose FSR is 1.5 GHz was chosen to measure the laser mode. The typical multi-mode operation is shown in Figure 4. The PZT's driving voltage of the Fabry-Perot interferometer is expressed by the yellow line \"voltage.\" Many peaks were observed in a free spectral range of 1.5 GHz, suggesting that the laser was operating in multi-mode. In order to realize the single-longitudinal mode of operation, two Fabry-Perot etalons with different thicknesses (1 mm and 6 mm) were inserted into the laser cavity. The thinner etalon was used to suppress other mode oscillations, and the thicker etalon was used to compress the laser line width even further. The SLM operation can be realized by optimizing the angle of the Fabry-Perot etalons. Figure 4: Output mode of the free-running Ho:SSO laser. Figure 3: The performance of the multi-mode Ho:SSO laser. Figure 5 shows that the Ho:SSO laser operates in an SLM. The voltage of the free spectral range was about 20 V. There were only two peaks in the 1.5 GHz free spectral range. Figure 6 shows the laser performance of the SLM Ho:SSO laser. In the experiment, a power meter was used for measuring the output power of the Ho:SSO laser. When the incident power was 10.2 W, we obtained a maximum output power of 590 mW at 2111.91 nm. At the same time, the slope efficiency and the optical conversion efficiency were 9.7% and 5.8%, respectively. The laser spectrum was recorded by the spectrum analyzer (0.7 pm resolution) when the incident pump power was 10.2 W. The central wavelength was 2111.91 nm with an FWHM of 0.11 nm. The result is shown in Figure 7.  
The beam quality factor \(M^{2}\) of the SLM Ho:SSO laser, using the travelling knife-edge method, was measured at 2111.91 nm, where the output power was the highest. The result can be seen in Figure 2.0 The data were fitted according to the Gaussian beam propagation equation and \(M^{2}\) was calculated to be 1.17. Figure 2.1 Output mode of the SLM Ho:SSO laser.