## 3 Results
### A Statistic Method to Evaluate the Threshold and Investigate the Threshold Conditions  
In order to determine the threshold of peak-power intensity and energy density for the initial breakdown under continuous-firing operation at a repetition rate of 0.2 Hz after warm-up, hundreds of mode-locked Q-switched (ML-QS) pulses were tightly focused into the atmospheric air. Note that the temporal profiles of these laser pulses have been monitored pulse-by-pulse to determine the region of threshold fairly. The pulse width of the envelope of ML-QS pulses was 7 ns on average. Figure 3a shows the relation of the successes of air breakdowns to the laser energy and laser peak power with ML-QS laser. The experimental results indicate that the success of the initial breakdown is slightly related to the pulse energy but mainly dominated by the peak power in the energy range from 2.10 to 3.05 mJ, corresponding to an energy density range from 1.36 to 1.98 kJ / cm\({}^{2}\). To analyze the influence of peak power on air breakdown for the specific laser cavity more explicitly, the statistics on the breakdown probability, \(\rho\), with respect to the peak power, \(P\), was performed in Figure 3b. For determining the threshold peak power, \(P_{th}\), threshold width of peak power, \(\Delta P\), and normalized threshold width of peak power, \(\Delta\widetilde{P}\), a phenomenological formula was used to fit the experimental data numerically as  
\[\rho(x)=\frac{1}{1+\exp\left[\frac{-(x-x_{th})}{\Delta x}\right]}\,\ x:\ P,\ E,\ I,ED \tag{1}\]  
\[\Delta\widetilde{x}=\frac{\Delta x}{x_{th}} \tag{2}\]  
where the \(x\) represents the evaluated laser parameter such as peak power, \(P\); pulse energy, \(E\); and peak-power intensity (peak power per unit area), \(I\); and pulse energy density (pulse energy per unit area), \(ED\). The solid red line shows the fitting curve given by (1) in Figure 3b,c,e,f. According to the numerical fitting, the \(P_{th}\) corresponding to 50% breakdown probability was 0.972 MW. In other words, the \(I_{th}\) was determined to be 0.63 TW/cm\({}^{2}\). In addition, the \(\Delta P\) and \(\Delta\widetilde{P}\) was found to be 44 kW and 0.045, respectively. Figure 2: (**a–f**) Mode-locked Q-switched pulses with various temporal profiles in the test energy range from 2.9 to 3.4 mJ. The histogram of breakdown probability with respect to the pulse energy was performed in Figure 3c. According to the numerical fitting, the \(E_{th}\), \(ED_{th}\), \(\Delta E\), and \(\Delta\widetilde{E}\) was found to be 2.35 mJ, 1.53 kJ/cm\({}^{2}\), 0.273 mJ, and 0.116, respectively. The \(\Delta\widetilde{E}\) was significantly larger than the \(\Delta\widetilde{P}\), which indicated the peak power was the principal threshold condition and consistent with the qualitative observation. In order to verify that the ML-QS pulse has a lower energy threshold than pure Q-switched (p-QS) pulse, the typical p-QS pulses with 7-ns duration were generated by adding a feedback mirror to inject a weak etalon effect. The p-QS laser pulse had almost the same beam quality and pulse width as the ML-QS laser pulses. Figure 3d-f show the experimental results and statistic analysis of air breakdown with p-QS pulses. According to the numerical fitting, the \(P_{th}\), \(I_{th}\), \(\Delta P\), and \(\Delta\widetilde{P}\) was found to be 1.07 MW, 0.70 TW / cm\({}^{2}\), 52 kW, and 0.048, respectively. The \(E_{th}\), \(ED_{th}\), \(\Delta E\), and \(\Delta\widetilde{E}\) was found to be 6.52 mJ, 4.23 kJ/cm\({}^{2}\), 0.382 mJ, and 0.059, respectively. The results demonstrate that the threshold peak power of ML-QS pulse was similar, only 0.91 times lower than the p-QS pulse. Moreover, the reason might be that the transverse mode quality of the ML-QS laser was still slightly better than the p-QS laser with feedback injection. However, the threshold energy of the ML-QS pulse was significantly 0.36 times lower than the p-QS pulse.