## 3 Experimental Setup and Theoretical Fitting  
To verify the model, we set up a concave-plano cavity for the PQS operation, and the experimental scheme is shown in Figure 1. The pumping source was a fiber-coupled laser diode with a core diameter of 0.2 mm, which was able to offer a maximum power of 40 W at 808 nm. The pumping energy from the laser diode was configured as pulse-pumping with the frequency 10 kHz and duty cycle 30% for reducing the thermal lensing effect. The input side of the resonator was a concave mirror with a radius of curvature 30 mm, and its entrance facet was coated anti-reflection films at 808 nm with reflectivity less than 0.1%; the second facet was coated high-reflection films at 1064 nm with reflectivity 99.8% and transmittance 96% at 808 nm. Since our model was based on the condition of no gain saturation, we chose a low-doped concentration 0.3 at. % Nd:YVO\({}_{4}\) as the gain medium to avoid saturation cases. Furthermore, the gain medium was an a-cut rectangular crystal with the dimensions of \(3\times 3\times 6\) mm\({}^{3}\), and both ends were coated anti-reflection films at 1064 nm with reflectivity less than 0.1%. The saturable absorbers were two Cr\({}^{4+}\):YAG with the initial transmission \(T_{0}=35\%\) and 50%, and both were rectangular crystals with the dimensions of \(3\times 3\times 3\) mm\({}^{3}\). The coating for the absorbers on both ends were the same as those of the gain medium. The gain medium and the saturable absorbers were wrapped with indium foils and mounted in copper holders that were connected to a water-cooling system keeping a temperature of 22 \({}^{\circ}\)C. The indium foils were used as high-efficiency heat sinks and filling material for tiny gaps between the crystal and the holder. Several flat mirrors with different reflectivity \(R_{oc}\) in a range from 10% to 92% were employed as the output couplers in this experiment. The distance \(d_{c}\) in Figure 1 is the average distance from the output coupler to the gain crystal, and \(d_{s}\) is the distance from the output coupler to the absorber. The cavity length was set to be 29 mm to satisfy the criterion of the second threshold condition. By \(w(d)=w_{0}\sqrt{1+\left(\lambda d/\pi w_{0}^{2}\right)^{2}}\), the mode sizes of the two ends and the middle point in the gain crystal can be evaluated to be 0.180 mm\({}^{2}\), 0.226 mm\({}^{2}\), and 0.202 mm\({}^{2}\) by substituting the \(d=28\) mm, 22 mm, and 25 mm, respectively. In modeling, we assumed the variation of the mode size in the crystal is small, then we ignored the effect of diffraction inside the gain crystal. Since the mode size in the middle is very close to the average mode size of the two ends, the approximation still holds in this case, so here we take \(w_{c}=0.226\) mm\({}^{2}\). By the experimental conditions that \(d_{s}=2\) mm, the mode sizes \(w_{s}\) in the saturable absorber can be obtained to be 0.046 mm. Adding the spontaneous emission coefficient of the gain crystal \(\sigma=15.6\times 10^{-19}\) cm\({}^{2}\) and the absorption coefficient of the ground state of the absorber \(\sigma_{gs}=8.7\times 10^{-19}\) cm\({}^{2}\), the value of \(\alpha\) can be evaluated to be \(\alpha=11\). To investigate the influence of the spatial overlap effect on the output energy, two sets of focusing lens were adopted for different scales of beam expansion. Consequently, the average pump beam radii of 0.1 mm and 0.3 mm were reimaged into the gain crystal with 90% coupling efficiency, respectively. Given the following experimental parameters \(d_{c}=25\) mm, \(l_{g}=6\) mm, \(z_{0}=25\) mm, \(\lambda_{p}=808\) nm, \(\delta=0.12\) mm\({}^{-1}\), \(n_{r}=2.22\), \(w_{c}=0.202\) mm, the integration in Equation (13) can be evaluated numerically, then substitute the result into the Equations (11) and (12) to obtain the effective mode area and overlap efficiency \(A_{eff}=0.344\) mm\({}^{2}\) and \(S=0.292\) for the pump size \(w_{p}=0.1\) mm, \(A_{eff}=2.883\) mm\({}^{2}\), and \(S=0.264\) for \(w_{p}=0.3\) mm.