## 1 Introduction  
Gas breakdowns generated by high-power optical pulses can be traced back to the groundbreaking era for lasers [1; 2]. Since then, the technologies of laser-induced breakdowns (LIBs) have been intensely studied and developed to be mature applications, including laser ablation [3], micromachining [4], photochemistry [5], laser fusion [6], laser-induced breakdown spectroscopy [7], laser ignition [8; 9], and cavitation bubble generation [10]. The LIB is generally described as the formation of ionized gas or plasma during or by the end of the excitation of the laser pulse. Physically, the breakdown and ionization of the gas molecules are triggered when the electric field of the laser pulse is greater than that of the binding electrons to their nuclei [11]. After the initial breakdown, the avalanche ionization through some cascade effects takes place to vastly increase the plasma density of the gas molecules [12; 13]. In addition to the gas breakdown, the interaction between ns laser and matter was deeply studied [14; 15]. Since LIB belongs to a threshold-like process with respect to the peak intensity of the laser pulse [16; 17; 18], determining the breakdown threshold has long been a critical issue for reliable results of the developed applications. In general, the threshold criterion for LIB is defined by the minimum laser energies at which 10-50% of the laser firings result in an observable optical breakdown [11; 19; 20; 21]. A few reports also propose that it is adequate to determine the breakdown threshold by the laser power intensity for taking the focused laser beam radius into account [18; 22; 23]. The thresholds of LIBs under different experimental conditions--such as laser wavelengths [24], pulse durations [25], gas pressures [26], compositions of gas molecules [27], and focal point properties [28, 29]--have been widely investigated. Due to the limitation of the response time of photodetectors, the temporal shapes of laser pulses were usually assumed to have Gaussian-like profiles for calculating the breakdown intensities in most works of literature [19, 20]. We call these laser pulses the pure Q-switched laser pulses. However, most of the temporal shapes of pure Q-switched pulse still have some small sub-pulse modulations by mode beating or slightly mode-locking that might not have been noticed for LIB in the past. In recent years, researchers have started to study the science of air-breakdown plasma using such modulated Q-switched pulse [23]. Moreover, because the actual temporal profiles of laser were not explicitly resolved and monitored pulse-by-pulse, the determined threshold intensities may have large fluctuations even under similar experimental arrangements [30, 31]. Thanks to the development of high-speed optical detecting technologies, the high-resolution temporal behaviors of laser pulses can be precisely measured. This improvement enables us to resolve the temporal dynamics further to find out the most crucial threshold condition for the initial breakdown. In this work, we utilized the lamp-pumped high-energy Q-switched Nd:YAG laser to explore the threshold conditions of the air breakdown. Based on the designed convex-concave resonator for good beam quality [32] and wedged gain crystal to avoid etalon effect for simultaneously mode-locking [33, 34]. The mode-locking modulation may have suffered from the instabilities caused by the inhomogeneous thermal loading and cavity misalignment [35, 36]. By utilizing this feature, we can generate laser pulses with various temporal shapes to study the influences of peak-power intensity on the gas breakdown. Both qualitative observation and the statistics results indicated that the breakdown threshold was determined mainly by the peak-power intensity instead of the pulse energy density. The threshold of peak-power intensity for air breakdown was found to be 0.63 TW / cm\({}^{2}\). The threshold width was also determined through a phenomenological formula.