## 4 Results and Discussion
### Pulse Width Evaluation  
Figure 3 shows the dependence of the signal intensities measured as a function of the prism displacement, \(X\), for laser beams emitted at 9\(\upomega\) (=267 nm), 10\(\upomega\) (=241 nm), and 11\(\upomega\) (=219 nm). Because \(\alpha X_{1/2}\) = 4.9 mm at 9\(\upomega\), the pulse width, \(\Delta t_{\mathrm{REAL}}\), can be calculated to be 64 fs from the data shown in Figure 2. For the values of \(\alpha X_{1/2}\) = 5.6 mm at 10\(\upomega\) and 1.7 mm at 11\(\upomega\), the corresponding values of \(\Delta t_{\mathrm{REAL}}\) can be calculated to be 78 and 49 fs, respectively. The spectral bandwidth was measured using a spectrometer, and the estimated values were _ca._ 3.3, 2.9, and 2.5 nm after calibration of the resolution (1.5 nm) of the spectrometer at 9\(\upomega\), 10\(\upomega\), and 11\(\upomega\), respectively. From the relationship of \(\Delta t\times\Delta\
u\geq k\) (where \(k\) = 0.441 for a Gaussian pulse), \(\Delta t_{\mathrm{FIL}}\) can be calculated to be 36, 34, and 35 fs at 9\(\upomega\), 10\(\upomega\), and 11\(\upomega\), respectively. The parameter, \(\upalpha\), can then be calculated to _ca._ 3.2, 5.3, and 2.0 for 9\(\upomega\), 10\(\upomega\), and 11\(\upomega\)  
Figure 2: Calculated dependence of the parameter, \(\Delta t_{\mathrm{REAL}}\), on the parameter, \(\alpha X_{1/2}\). The laser frequency, n\(\upomega\), is shown in the figure, where n and \(\upomega\) are the order of Raman sidebands and the Raman shift frequency of molecular hydrogen (\(\upomega\) = 4155 cm\({}^{-1}\)), respectively. The definitions of the parameters, \(\Delta t_{\mathrm{REAL}}\) and \(\alpha X_{1/2}\), are given in the text. respectively. These results suggest that the anti-Stokes beam has the shortest pulse width among them and has a slightly (1.4 times) larger pulse width than the width expected for a transform-limited pulse. While the spectral profile of the three-color beam composed from the three spectral lines (267, 241, 219 nm) was far beyond the Gaussian shape, the presented technique was applied to this type of beam on a trial basis. The predicted results were calculated for two cases: (1) where the phases of the emissions are random; and (2) where the emission lines are phase-locked. In the former case, a broad band profile was obtained, as shown in Figure 4a, in which \(\alpha X_{1/2}\) = 2.8 mm. In the latter case, a very sharp peak, _i.e._, \(\alpha X_{1/2}\) = 0.015 mm, was obtained, as shown in Figure 4b. The experimental data shown in Figure 4c consisted of two components, where one is a sharp peak (\(\alpha X_{1/2}\) = 0.12 mm) at the center, and the other is a broad band observed as a pedestal. It should be noted that the signal intensity was highly sensitive to the positions of the second prisms in the vicinity of the maximum value, and that this result was obtained by carefully translating these prisms simultaneously after critical optimization of their positions. It is possible to consider that a sharp peak would arise from the phase-locked components while another would arise from the phase-random components. The former peak width (0.12 mm), corresponding to a pulse width of 11 fs, was apparently broader than the theoretically predicted value (0.015 mm), corresponding to a pulse width of 4.0 fs. This discrepancy is likely to arise from insufficient precision in phase control of the three emissions, during which the prisms were moved simultaneously using translation stages equipped with manually driven differential micrometers (see Figure 1b). Another explanation for the discrepancy, and the appearance of the pedestal, could be the spectral phase and amplitude fluctuations between the three pulses used in this study [32]. Figure 3: Observed dependence of the signal intensity, \(\hat{I}^{2}\)\({}_{\text{FRAC}}\) (\(\tau=0\)), on the parameter, \(X\). The dots in the figure are the observed data, and a solid curve is a guide to the eye. The parameter of \(X_{1/2}\) can be obtained by calculating the half width at the half maximum of the signal peak.