## 3 Results and Discussion
The solid lines in both figures are the results from the measured multiphoton absorption (MPA) parameters in Reference [17], which are clearly indicated in the legends, respectively. when propagation distance increases (Figure 4b,e). However, at the end of filament formation, the peak fluence in dry air becomes smaller than in wet air (Figure 4c,f). Relatively, the radius of filament is wider but its intensity is lower in wet air when compared with in dry air. Figure 5a-d details the temporal distortions of the pulse at four different propagation distances. During the self-focusing stage, the pulse keeps its initial Gaussian shape and undergoes only an increase of its peak intensity (Figure 5a). The peak value in dry air is much higher than in wet air Obvious multi-peaked structures and pulse broadening can be observed during the filamentation process, as noted in Figure 5b,c. The trailing pulse and leading peak in temporal profile is created by plasma defocusing [10; 22; 23]. The symmetric spikes of the pulse in time are associated with the GVD [24]. The GVD can further broaden the temporal width [14]. As shown in in Figure 5b,c, the number of splits in temporal profile changes little in both types of air gas. The degree of compression of a femtosecond pulse depends on air humidity. The broadening of laser pulses in dry air is much more obvious than in wet air. In the end of filamentation, the pulse propagated in wet air broadens and its peak intensity exceeds that in dry air (Figure 5d). Figure 4: Fluence distribution F/F\({}_{0}\)(x,y) at different propagation distance z. The top row, (**a**) z = 2.0 m, (**b**) z = 3.5 m, and (**c**) z = 7.0 m for dry case. The bottom row, (**d**) z = 2.0 m, (**e**) z = 2.5 m, and (**f**) z = 7.0 m for wet case. Figure 5: _Cont._  
In meteorology, relative humidity (RH) is an important physical measure to describe the amount of water vapor present in ambient conditions. We further investigated the nonlinear propagation of UV pulses in conditions with different relative humidity. The relative humidity grows as the amount of water vapor increases. By definition, relative humidity can be expressed as the ratio of vapor pressure (e) to saturation vapor pressure (\(e_{sw}(T)\)):  
\[RH=\left[\frac{e}{e_{sw}(T)}\right]_{p,T} \tag{4}\]  
The saturation vapor pressure of pure water is only a function of temperature, which can be calculated according to [PERSON]'s formulations [25] as follows:  
\[ \begin{array}{ll}e_{sw}=&0.01\exp\left[-2991.2729T^{-2}\right.\\ &\left.-6017.0128T^{-1}+18.87643854\right.\\ &\left.-0.028354721T+0.17838301\times 10^{-4}T^{2}\right.\\ &\left.-0.84150417\times 10^{-9}T^{3}+0.44412543\times 10^{-12}T^{4}\right.\\ &\left.+2.858487\ln T\right]\end{array}  \tag{5}\]  
where \(e_{sw}\) is in hPa and \(e_{0}\) = 6.11 hPa is the saturated vapor pressure at 0 \({}^{\circ}\)C. \(T\) is absolute temperature in K. For moist air, a slight correction must be made to calculate the saturation vapor pressure, which is expressed by:  
\[e^{\prime}_{sw}(p,T)=f(p)\times e_{sw}(T) \tag{6}\]  
where \(f(p)\) is called the enhancement factor and its formulation is given below [26]:  
\[f(p)=1.0016-0.074p^{-1}+3.15\times 10^{-6}p \tag{7}\]  
Based on Equations (5)-(7), the saturation vapor pressure in moist air is 24.991 hPa at T = 21 \({}^{\circ}\)C. The vapor pressure e can then be derived from the ideal gas state equation:  
\[pV=nRT \tag{8}\]  
Figure 5: Temporal profile as functions of the propagation distance z. (**a**) z = 2.0 m, (**b**) z = 2.75 m, (**c**) z = 4 m, and (**d**) z = 5.5 m. The blue and green solid lines represent results from the measured MPA parameters in dry and wet air conditions, respectively. The dry air is assumed to only consist of N\({}_{2}\) and O\({}_{2}\) in a ratio of 0.78 to 0.21. For wet air, we impose a composition of 78.0% N\({}_{2}\), 21.0% O\({}_{2}\), and 1.0% H\({}_{2}\)O (gas). The black dash line represents the simulated result from the calculated MPA parameters in Reference [9].