## 2 Experimental Setup  
The experimental setup is illustrated in Figure 1. In the experiment, the concentration of Tm ions is 4 atom %. The size of the crystal is 3 \(\times\) 3 \(\times\) 15 mm. Both of the facets are anti-reflection (AR) coated at 792 nm (\(R\) < 0.5%) and 1.94 \(\upmu\)m (\(R\) < 0.3%). In order to keep Tm:YAP crystal at 18 \({}^{\circ}\)C, a thermoelectric cooler (TEC) was used to precisely control the temperature of the Tm:YAP crystal, which was placed in a copper heat-sink and wrapped in indium foil. The pump source was a laser diode whose maximum output power was 60 W, and the core-diameter of the pigtail fiber was 400 \(\upmu\)m (Nlight, Vancouver, WA, US; P4-060 central wavelength: 793.2 nm; FWHM: 1.4 nm). The focal lengths of the two mode-matching lenses are 25 mm and 45 mm, respectively, in order to realize better mode matching. In the center of the laser crystal, the diameter of the pump-beam was nearly 720 \(\upmu\)m. The 45\({}^{\circ}\) dichroic mirror with high-transmittance (HT) coated at 792 nm (\(T\) \(\sim\) 95%) and high-reflectivity (HR) coated at 1.9 \(\upmu\)m (\(R\) > 99.5%) was used to deplete the pump light. The inserted Fabry-Perot etalons (1 mm and 6 mm) were used to shift the output wavelength to match the absorption spectrum of the Ho:SSO medium. A plano-concave lens was chosen as the output coupler whose radius of curvature was 300 mm.  
To keep the Ho:SSO crystal at 18 \({}^{\circ}\)C, a thermoelectric cooler (TEC) was used to precisely control the temperature of the Ho:SSO crystal, which was fixed on a copper heat-sink and wrapped in indium foil. In this experiment, the doping concentration of the Ho:SSO crystal is 0.5 atom %, and the size is 5 \(\times\) 5 \(\times\) 13.8 mm. Both of the end surfaces of the crystal are AR-coated (\(R\) < 0.3%) at 1.94 \(\upmu\)m. The actual single-absorption efficiency of the Ho:SSO crystal was measured to be 58%. A plano-concave resonator was utilized in the Ho:SSO laser. A plano-concave lens was chosen as an output coupler whose transmissions is 15% at 2.1 \(\upmu\)m and has a concave curvature radius of 200 mm. The 45\({}^{\circ}\) dichroic mirror is HR coated (\(R\) > 99.5%) at 1.94 \(\upmu\)m and HT (\(T\) > 94%) at 1.94 \(\upmu\)m. Near the center of the laser crystal, the diameter of the pump beam was about 480 \(\upmu\)m. The transmission of both flat surfaces is 99.2% at 2.1 \(\upmu\)m by AR coated with SiO\({}_{2}\) infrasil. In order to achieve SLM laser output, two F-P etalons (1 mm and 6 mm) made by YAG materials with 20% reflection loss were used. According to the following expressions, \(\Delta v_{c}=c/2 nd\), where \(\Delta v\) is the free spectral range (FSR), and \(d\) is the thickness of the etalon. The FSR of the 1 mm and 6 mm etalons are calculated to be 8.24 \(\times 10^{10}\) Hz and 1.37 \(\times\) 10\({}^{10}\) Hz, respectively. The mode selection effect of the 6 mm etalon has a narrower transmission curve, which is better than that of the 1 mm etalon. Thus, the 1 mm etalon was inserted first, and then the 6 mmone. By adjusting the angle of the two etalons, the other modes were suppressed due to high losses, and only one mode achieved the highest gain in the peak of the transmittance.