## 4 Results and Discussions  
Figure 2 shows the plot of the average output power versus the input power in the pulse pumping scheme for \(T_{0}=35\%\) and 50% with the output coupler \(R_{oc}=40\%\). The results show a characteristic staircase structure where the plateaus indicate the locking effect at various simple frequency ratio. This phenomenon is consistent with the previous results [19].  
Figure 3 shows a typical stable pulse train and the pulse width of a single pulse with time span division of 8 ms and 40 ns in the case of \(T_{0}=35\%\) to the frequency-locked states of 1:1, i.e., \(f_{\text{rep}}=10\) KHz. The overall pulse-to-pulse amplitude fluctuation was found to be within 5%. The pulse duration was measured approximately to be 2 ns, as shown in Figure 3b. We also observed that there was no additional pulse or satellite pulse in every single Q-switched shot.  
The output energy versus the output coupler reflectivity of the theoretical fitting and the experimental results with \(T_{0}=35\%\) for two pumping foci are shown in Figure 4a. It can be clearly seen that the values of the curve of the case \(w_{p}=0.3\) mm is higher than that of the case \(w_{p}=0.1\) mm by a factor of 1.25 on average, which is exactly corresponding to the ratio of the energy factor 65 \(\upmu\)J to the energy factor 52 \(\upmu\)J. The results for other cases of \(T_{0}=50\%\) is also shown in Figure 4b. The agreement of the theoretical calculations and the experimental data confirms the validity of the developed spatial model.  
Figure 3: (**a**) A stable pulse train with fluctuation deviation less than 5%. (**b**) A single pulse of 2 nanoseconds generated by a PQS Nd:YVO\({}_{4}\)/Cr\({}^{4+}\):YAG laser.  
Figure 2: The plot of the output power versus the input power of two pumping foci with the initial transmission (**a**) 35% and (**b**) 50% of the saturable absorbers.  
By the condition \(d\Phi/dN=0\) from Equation (12), the inversion population when the maximum photon number occurs is almost equal to \(N_{t}\) in our cases with \(\alpha=11\). Thus, the spatially dependent peak power of the PQS model can be evaluated by  
\[P_{peak}=\frac{\hbar v_{l}}{t_{r}}(1-R_{oc})\Phi(N_{t}) \tag{23}\]  
and the space-related terms \(S\) and \(V_{eff}\) are included in the maximum photon number \(\Phi(N_{t})\). The theoretical fitting and experimental results for the peak powers are shown in the Figure 5 for the cases \(w_{p}=0.1\) mm and \(w_{p}=0.3\) mm with \(T_{0}=35\%\). The peak powers for both pumping foci possess the similar pattern to that of the output energies owing to the spatial overlap effect too. From that, again, we see the good agreement between the theoretical fitting and experimental results.  
The above result and discussion are based on the ratios of the output frequency to the input frequency to be 1:1, which means that the input energy during one pulse-pumping cycle is enough to generate a corresponding lasing pulse. From the experimental results, we find that the model is still valid at various simple frequency ratios such as 1:2 and 2:3, which is the frequency locking effect as the first and second plateau shown in Figure 2. The time-domain pulse train is shown in Figure 6 for the case that \(w_{p}=0.1\) mm, \(T_{0}=35\%\), \(R_{oc}=40\%\) with pulse-pumping frequency 10 kHz, and the output energy factor \(\hbar v_{1}A_{eff}S/(2\sigma\gamma)\) for the pump size is still around 52 uJ.  
Figure 4: Theoretical fitting and experimental results of the output energy of the two pumping foci \(w_{p}=0.1\) mm and \(w_{p}=0.3\) mm with (**a**) \(T_{0}=35\%\) and (**b**) \(T_{0}=50\%\).  
Figure 5: Theoretical fitting and experimental results of the peak power of the two pumping foci \(w_{p}=0.1\) mm and \(w_{p}=0.3\) mm with (**a**) \(T_{0}=35\%\) and (**b**) \(T_{0}=50\%\).