## 3 Results and Discussion
### Relative Phases among Raman Sidebands  
The Raman sidebands can be generated under phase-matched conditions in four-wave mixing [23]. It should be noted that the relative phases among these emission lines plays an important role in the temporal shape of the synthesized pulse. Figure 5A shows the temporal shape of the pulse calculated under the assumption that the relative phase is zero (ideal case). A train of 2.6-fs pulses can be observed with a spacing of 8 fs in the envelope of the laser pulse (35 fs). On the other hand, the data calculated under the assumption that the relative phase is shifted by \(\pi\), 0, 0 for the beams emitting at 240, 267, and  
Figure 4: Temporal profile of the pulse train (black curve) and the phase of the pulse (blue curve).  
Figure 3: FROG trace. (**A**) Original data (**B**) retrieved data. A fused silica plate (10 μm) was used as the SD material.  
300 nm is shown in Figure 5B. A train of 2.6-fs pulses with a spacing of 4 fs is generated. These data suggest that controlling the relative phases of the emissions is desirable for generating short optical pulses with a sufficient spacing, as mentioned in the following section.  
Figure 6 shows the calculated FROG trace, in which three transform-limited pulses (240, 267, 300 nm) are assumed to have equal intensities. Seven components appear in the trace, and five, which appear at 342, 300, 267, 240, 219 nm, have fringe patterns separated by 8 fs. Table 1 shows the origins for these seven components.  
Note that the components appearing at 200 and 400 nm are generated with a single pair of weak components emitting at 240/240 nm and 300/300 nm, respectively. As a result, only the weak component with no fringe pattern was observed in the trace. These weak signals would likely be buried in the noise, when the intensities of the Raman sidebands are relatively small against the intensity of the probe beam. It should be noted that the SD FROG signal appears through a third-order, nonlinear process, and decreases significantly at low emission intensities. For the other five components, more than two origins contribute to the appearance of the components, which explains the generation of fringe patterns. Note that at least two components emitting at 267 nm are required for the generation of the SD signal. Thus, these components were clearly observed in the trace. In order to generate the fringe pattern, the intensity of the pulses emitting at 240, 267, and 300 nm should be nearly equal. In this study, the probe pulse (267 nm) was attenuated by replacing the pair of aluminum mirrors used in the roof mirror with gold mirrors, which have a lower reflectivity in the DUV region. The relative intensity distribution was measured to be 45:100:53 for the emissions at 240, 267, and 300 nm using the spectrometer, which is nearly ideal for a Fourier synthesis of the emissions to generate short optical pulses.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Component** & **Origin** & **Component** & **Origin** \\ \hline
**Wavelength (nm)** & **(Wavelength, nm)** & **Wavelength (nm)** & **(Wavelength, nm)** \\ \hline
400 & \(300+300-240\) & 267 & 267 + 240 \(-240\) \\
342 & \(300+300-267\) & - & 240 + 300 \(-267\) \\ - & \(300+267-240\) & 240 & 300 + 240 \(-300\) \\
300 & \(300+300-300\) & - & 267 + 267 \(-300\) \\ - & \(