## 2 Theoretical Calculations
The actual pulse width is, however, broadened by several reasons: (1) an initial laser pulse would be neither Gaussian-shaped nor transform-limited; (2) the laser pulse would be chirped even at the optimal prism position because of the third-order dispersion (TOD) of the fused silica of the prisms used for pulse compression; and (3) the pulse front would be deformed by mode-change and self-focusing during beam transmission in a hollow capillary filled with hydrogen gas (many hot spots were observed in the beam pattern). In order to take account of the deterioration of the laser beam, a parameter, \(\alpha\), can be introduced into Equation (12). \[I(t)=\left|E_{0}\right|^{2}e^{-2 at^{2}} \tag{2}\]2  
where \(\Delta t_{\text{REAL}}\) is the pulse width observed in the experiment and \(\alpha\) is the parameter showing the deviation from the transform-limited pulse. When TOD is only a factor responsible for the distortion of the pulse, \(\alpha\) can be written as \(1+TOD/GVD\) where  
\[I(t)=\left|E_{0}\right|^{2}e^{-2 at^{2}} \tag{2}\]3  
Equations (11)-(13) lead to the following equations. \[I(t)=\left|E_{0}\right|^{2}e^{-2 at^{2}} \tag{2}\]4  
\[I(t)=\left|E_{0}\right|^{2}e^{-2 at^{2}} \tag{2}\]5  
As shown in Equation (16), the value of \(X_{1/2}\) to be obtained for a transform-limited pulse is actually expanded to a value of \(\alpha X_{1/2}\) in the experiment by some undesirable effects arising from additional chirps and deterioration of the laser pulse. The graph showing the relationship between \(\Delta t_{\text{REAL}}\) and \(\alpha X_{1/2}\) is the same as that of \(\Delta t_{\text{FTL}}\)_vs._\(X_{1/2}\), as shown in Equations (15) and (16). As can be recognized from Equations (15) and (16), \(\Delta t_{\text{REAL}}\) is expanded by a factor of \(\sqrt{\alpha}\) from \(\Delta t_{\text{FTL}}\) that can be  
\[I(t)=\left|E_{0}\right|^{2}e^{-2 at^{2}} \tag{2}\]6 **(a)** Prism parameters; **(b)** configuration of the prisms in a pulse compressor. Dotted arrows show the direction of displacement of the prisms. The parameters, \(\ell\) and \(L\), are the path length of the laser beam in the prism and the distance from the top of the prism to the laser beam, respectively. The angle of the prism is specified in the figure. Three prisms in **(b)** can be manually translated by rotating the nobs of the differential micrometers independently. calculated from the spectral bandwidth of the laser beam. The parameter, \(\alpha\), can be calculated from the ratio of \(\Delta t_{\rm REAL}\) and \(\Delta t_{\rm FTL}\), suggesting a degree of the deviation from the ideal transform-limited Gaussian pulse. A train of ultrashort optical pulses is generated by superposition of the laser emissions, which are phase-locked to each other. In this study, three beams, _i.e._, 9\(\omega\) (267 nm), 10\(\omega\) (241 nm), and 11\(\omega\) (219 nm), where \(\omega\) = 4155 cm\({}^{-1}\), were superimposed and were then used as one of the test beams. The spectral width, \(\Delta\omega_{\rm n}\), and the parameter, \(a_{n}\), where \(n\) = 1, 2, 3, for each beam, were assumed to be \(\Delta\omega_{1}\) = \(\Delta\omega_{2}\) = \(\Delta\omega_{3}\) = \(\Delta\omega\) and \(a_{1}\) = \(a_{2}\) = \(a_{3}\) = \(a\) for the purposes of this study. A train of ultrashort pulses is formed under these conditions if the three emission lines are phase locked. Thus, the phase locking expected to occur during the process of four-wave Raman mixing, can be confirmed by comparing the experimental data with the simulation results. The GDD can then be expressed as follows [2; 21].